## wpiea2026190

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### I. Introduction — Intelligence Divide and Human Capital
- Central research question: Will artificial intelligence (AI) help poorer countries catch up, or widen existing gaps by lowering the cost of using knowledge while favoring economies with existing skilled workers, reliable data, capable firms, and effective institutions?
- Key conceptual distinction:
  - Intelligence Divide: unequal capacity to convert increasingly cheap cognitive services into sustained productivity gains (not merely unequal access).
  - Empirical counterpart: productivity mobility — whether economies leave low-productivity states and move toward the frontier.
- Theoretical framing links appropriate-technology and absorptive-capacity literatures to a distribution-dynamics empirical design.
- Operational design highlights:
  - Finite-state Markov framework (Quah-style) to study transition probabilities across relative-productivity states.
  - Caselli–Coleman decomposition to separate level effects (workforce composition) from mobility effects (probability of moving toward frontier).
- Core empirical pattern (fixed 1975 non-U.S. quartiles, 1,097 transitions):
  - Mean time required to leave the lowest state:
    - capital intensity: 16.1 years
    - measured human capital: 31.5 years
    - TFP: 44.0 years
  - Interpretation: observable inputs accumulate faster than they are translated into sustained relative-productivity improvements.
- Human capital and mobility (median split):
  - Mean exit time from the lowest-productivity state for observations below the within-year median: 79.1 years
  - Mean exit time for observations above the within-year median: 28.0 years

### II. Methods — Distribution Dynamics, Decomposition, and Data
- Markov framework:
  - Discrete-time process {X_t} with finite state space S={1,...,K}; transition matrix P = [p_ij]; transitions measured over five-year intervals.
  - Variables expressed relative to the United States: x̃ = x / x_{US,t}; United States used as benchmark frontier.
  - Reported objects: transition probabilities, mean exit times, mean first-passage times, convergence statistic γ = − log_2 ( log |λ_2| ) where λ_2 is second-largest eigenvalue; γ measured in transition periods (multiply by interval length for years).
  - Stationary distribution ψ satisfies ψ = ψP; used as conditional long-run summary.
- Caselli–Coleman decomposition (production-accounting specification):
  - Y_i = K_i^α (A_i E_i)^{1−α}, α = one-third.
  - Effective labor E_i = [ (A_{u,i} L_{u,i})^{(σ−1)/σ} + (A_{s,i} L_{s,i})^{(σ−1)/σ} ]^{σ/(σ−1)}.
  - Decomposition: log(TFP_i / TFP_US) = log(A_{u,i} / A_{u,US}) + log(B_i / B_US) + ε_i.
  - Interpretation: first term captures relative-efficiency/organizational margins; log(B/B_US) is workforce-composition term under common skill-premium assumptions; ε_i is residual.
- Data sources and samples:
  - Penn World Table 11.0 (PWT 11.0): main production sample over 1960–2015 in non-overlapping five-year intervals.
  - Maddison Project: long-run relative income.
  - Barro-Lee educational-attainment data (through 2015): skilled/unskilled shares for decomposition.
  - Coverage: pooled decomposition 120 countries, 1,186 country-period observations; baseline relative-TFP transition matrix uses 1,097 five-year transitions; each country-period transition equally weighted.
- Estimation and inference:
  - Transition probabilities are sample-frequency MLEs; Anderson–Goodman standard errors reported; country-block bootstrap used for threshold inference (300 replications).
- Noted data limitations:
  - Educational attainment imperfect proxy for capability; international productivity comparisons sensitive to PPP, national-accounts quality, and TFP construction.

### III. Key Empirical Findings on Persistence and Mobility
- Long-run persistence (five-year transitions, states defined from 1975 cross-section):
  - Economies in the lowest state remain there with probability 0.943 over five years (Maddison Project data).
  - Economies in the highest state remain there with probability 0.931 over five years.
- Mobility characteristics:
  - Movements are mostly local; jumps from the bottom to the top are absent.
  - Lower tail is durable and not merely a cross-section artifact.
- Appendix A development-times:
  - Mean exit time from the lowest income state: 87.7 years.
  - Mean first-passage time from the bottom to the top: 330.8 years.
- Transition-pattern summaries (capital, human capital, relative TFP; five-year probabilities and implied durations):
  - Capital-output ratios:
    - Five-year probability of leaving bottom state: 0.327
    - Mean exit Q1 (years): 15.3 (Table 2)
    - Stationary Q4 mass: 0.771
  - Human capital (PWT hc):
    - Bottom-state five-year upward-mobility probability: 0.156
    - Mean exit Q1 (years): 32.1
    - Stationary Q4 mass: 0.903
  - Relative TFP (PWT ctfp):
    - Lowest-state five-year persistence: 0.886 (upward movement 0.114)
    - Mean exit Q1 (years): 44.0
    - Stationary Q4 mass: 0.089
- Stricter comparison on the same 1,097 transitions (1975 non-U.S. quartiles):
  - Mean exit from Q1: capital intensity 16.1 years; human capital 31.5 years; TFP 44.0 years.
  - Upward mobility probabilities: 0.310 (capital), 0.159 (human capital), 0.114 (TFP).
- Relative-TFP five-year transition matrix (1960–2015, state thresholds: <0.65, <0.88, <1.04, >=1.04; N_transitions per row reported):
  - From <0.65 (N_transitions 449): 0.886, 0.111, 0.002, 0.000 (SEs: (0.015) (0.015) (0.002) (0.000))
  - From <0.88 (N_transitions 297): 0.185, 0.640, 0.155, 0.020 (SEs: (0.023) (0.028) (0.021) (0.008))
  - From <1.04 (N_transitions 207): 0.019, 0.271, 0.551, 0.159 (SEs: (0.010) (0.031) (0.035) (0.025))
  - From >=1.04 (N_transitions 144): 0.000, 0.056, 0.285, 0.660 (SEs: (0.000) (0.019) (0.038) (0.039))
  - Stationary distribution: 0.479, 0.278, 0.154, 0.089

### IV. Decomposing Relative Productivity — Caselli–Coleman Findings
- Decomposition results (2020 auxiliary cross-section, Table 4) by human-capital quartile:
  - Q1 (low HC), N 29:
    - Mean rel. TFP obs.: 0.486
    - log(TFP/TFP_US): -0.782
    - log(A_u / A_u_US): -2.469
    - log(B / B_US): 0.312
    - log residual: 1.375
  - Q2, N 29:
    - Mean rel. TFP obs.: 0.657
    - log(TFP/TFP_US): -0.484
    - log(A_u / A_u_US): -1.52
    - log(B / B_US): 0.371
    - log residual: 0.665
  - Q3, N 28:
    - Mean rel. TFP obs.: 0.727
    - log(TFP/TFP_US): -0.368
    - log(A_u / A_u_US): -0.991
    - log(B / B_US): 0.23
    - log residual: 0.393
  - Q4 (high HC), N 29:
    - Mean rel. TFP obs.: 0.759
    - log(TFP/TFP_US): -0.304
    - log(A_u / A_u_US): -0.671
    - log(B / B_US): 0.079
    - log residual: 0.288
- Visual and numerical conclusion: cross-country variation is much greater along the relative-efficiency axis than the workforce-composition axis; workforce composition contributes but does not dominate productivity differences.

### V. Human Capital and Mobility — Capability Regimes
- Endogenous threshold exercise (Hansen-style) selects cutoff τ̂ maximizing the likelihood-ratio:
  - Estimated threshold τ̂: 0.676
  - 95% bootstrap CI lower: 0.398
  - 95% bootstrap CI upper: 0.841
  - LR at τ̂: 60.57 (300 bootstrap replications)
  - Mean exit time from Q1 | HC < τ̂ (years): 64.8
  - Mean exit time from Q1 | HC ≥ τ̂ (years): 25.4
  - Exit-time ratio (Low/High): 2.55×
  - N transitions, HC < τ̂: 632
  - N transitions, HC ≥ τ̂: 476
- Operational cutoff used for comparisons: 0.64 (close to τ̂ and comparable to upper human-capital state boundary).
- Regime contrasts under operational split:
  - Low-capability regime:
    - Q1 persistence: 0.922
    - Upward movement over five years: 7.8 percent
    - Stationary distribution assigns 68.8 percent mass to Q1 and 5.2 percent to Q4
  - High-capability regime:
    - Q1 persistence: 0.826
    - Upward mobility: 17.4 percent
    - Stationary Q4 mass: 12.8 percent
  - Mean exit time from Q1 falls from 64.3 to 28.7 years; expected passage from Q1 to Q4 falls from 368.5 to 177.0 years.
- Regression-adjusted mobility (supporting checks):
  - Continuous human-capital coefficient for five-year growth in log relative TFP: 0.222 higher (remains 0.200 after controlling for income and capital intensity, p < 0.001).
  - Adding exact initial productivity reduces coefficient from 0.417 to 0.150 (p = 0.068); adding initial income reduces coefficient further to 0.028 (p = 0.790).
  - Fixed-rank and smooth specifications both improve out-of-sample transition prediction.

### VI. AI Counterfactuals — Skill Augmentation vs Diffusion
- Two comparative-static AI interpretations:
  1. Skill-augmenting technological change:
     - A_{s,i}' = (1 + θ) A_{s,i}, θ > 0.
     - Gains depend on existing supply of skilled labor and production systems; raises return to existing capabilities without changing capability-acquisition processes.
  2. Diffusion technology:
     - Lowers costs of codifying, translating, diagnosing, adapting, and applying knowledge; operates through changes in the transition matrix and can raise upward mobility in low-productivity economies.
- Accounting sensitivity exercise (Table 8) — AI as a skill-biased premium shock (entries: AI scenario / HC path / log B (High HC) / log B (Low HC) / Composition gap ∆B):
  - Baseline (π =1.80) Stagnant HC: "1.476" | "1.563" | "-0.088"
  - Baseline (π =1.80) Rising HC: "1.476" | "1.653" | "-0.177"
  - Moderate AI (π =2.50) Stagnant HC: "1.706" | "1.732" | "-0.025"
  - Moderate AI (π =2.50) Rising HC: "1.706" | "1.85"  | "-0.143"
  - Strong AI (π =3.00) Stagnant HC: "1.838" | "1.829" | "0.009"
  - Strong AI (π =3.00) Rising HC: "1.838" | "1.963" | "-0.125"
- Interpretation: as the AI premium rises, the composition gap can narrow or change sign under stagnant human capital; skill-biased AI does not mechanically cause convergence or divergence — outcomes depend on initial skill distribution and subsequent capability formation.
- Diffusion channel and practical implementation:
  - Diffusion can reduce costs of codifying tacit knowledge, translating technical information, diagnosing production problems, and adapting frontier practices.
  - Aggregate productivity gains from diffusion may be delayed or initially negative during installation because of required accumulation of data, training, workflow redesign, and organizational capital.
  - For follower economies, adaptation (fitting tools to local languages, data, regulations, production processes, service-delivery systems) is central.

### VII. Scenario Illustrations and Long-Run Projections
- Scenarios (conditional illustrations, not forecasts): Baseline (unconditional relative-TFP matrix), High-HC transition matrix, Broad AI diffusion (scale =1.0), Unequal AI diffusion (scale =1.0).
- Selected long-run comparative outcomes (Table 9; Horizon columns show Q1–Q4 shares, γ (years), MET1 (years), Passage Q1→Q4 (years)):
  - Baseline (unconditional TFP) 25 years:
    - Q1: "0.510" | Q2: "0.276" | Q3: "0.140" | Q4: "0.074"
    - γ (years): "21.2"
    - MET1 (years): "44.0"
    - Passage Q1→Q4 (years): "241.8"
  - Baseline (unconditional TFP) 100 years: Q1 "0.482" | Q4 "0.088" | γ "21.2" | MET1 "44.0" | Passage "241.8"
  - High-HC transition matrix 25 years: Q1 "0.356" | Q2 "0.360" | Q3 "0.206" | Q4 "0.077" | γ "20.6" | MET1 "28.7" | Passage "177.0"
  - High-HC transition matrix 100 years: Q1 "0.262" | Q4 "0.124" | MET1 "28.7" | Passage "177.0"
  - Broad AI diffusion (scale =1.0) 100 years: Q1 "0.340" | Q4 "0.105" | γ "17.2" | MET1 "28.6" | Passage "187.6"
  - Unequal AI diffusion (scale =1.0) 100 years: Q1 "0.545" | Q4 "0.087" | γ "25.2" | MET1 "58.8" | Passage "285.9"
- Interpretation:
  - Applying a high-capability transition matrix materially shortens lower-tail persistence but does not produce rapid convergence.
  - Broad AI diffusion can improve mobility meaningfully; unequal diffusion can worsen lower-tail persistence.

### VIII. Robustness, Stability, and Timing Sensitivity
- Transition-horizon (Chapman-Kolmogorov) check (Table 10):
  - Comparison: directly estimated ten-year matrix P₁₀ vs P₅² (two-step five-year):
    - γ (years): 5-year baseline "21.186" vs 10-year direct "12.102" | Difference "-9.084"
    - MET Q1 (years): "44.020" vs "29.923" | Difference "-14.097"
    - Passage Q1→Q4 (years): "241.825" vs "149.076" | Difference "-92.749"
    - Stationary Q1: "0.479" vs "0.478"
    - Stationary Q4: "0.089" vs "0.085"
    - ‖P₁₀ − P₅²‖F: "0.098"
    - Max |entry|: "0.041"
    - N transitions (10-year): "978"
  - Interpretation: direct ten-year matrix implies faster movement but similar long-run distribution; timing estimates are horizon-sensitive.
- Temporal stability (pre- and post-1995, Table 11):
  - Early period (<1995) vs Late period (≥1995):
    - Q1 persistence: "0.870" vs "0.911"
    - Upward mobility from Q1: "0.130" vs "0.089"
    - Mean exit time Q1 (years): "38.409" vs "56.190"
    - Passage time Q1→Q4 (years): "152.481" vs "452.799"
    - Stationary Q1 mass: "0.350" vs "0.612"
    - Stationary Q4 mass: "0.154" vs "0.036"
    - N transitions: "517" vs "476"
    - χ2 homogeneity test (p-value): "0.4500"
  - Interpretation: chi-squared test does not reject equality; point estimates show greater lower-tail persistence after 1995.
- Discretization and alternative measures:
  - Alternative discretizations (absolute thresholds [0.20,0.40,0.70], time-varying quartiles, terciles, quintiles) preserve qualitative result of lower-tail stickiness under fixed-rank baseline.
  - Cognitive-skills robustness (Hanushek–Woessmann, 69-country sample) shows similar regime contrasts:
    - Low cognitive skills (HW ≤ 4.69): P[Q1,Q1] = 0.881 (SE 0.028), Mean Exit Q1 = 41.9 years, Passage Q1→Q4 = 196.6 years.
    - High cognitive skills (HW > 4.69): P[Q1,Q1] = 0.818 (SE 0.047), Mean Exit Q1 = 27.5 years, Passage Q1→Q4 = 111.0 years.

### IX. Policy-Relevant Conclusions and Actionable Priorities
- Core insight: access to AI alone does not equate to capacity to benefit; the relevant policy question is whether AI relaxes capability constraints that limit productivity mobility.
- Complementary investments matter for converting inputs into sustained productivity improvements:
  - Education quality, management and organizational capacity, infrastructure, finance, state capability, reliable data, and institutions that enable experimentation and learning.
- For AI to promote convergence it must change the process by which economies acquire, adapt, and use productive knowledge (diffusion-enhancing AI), not merely raise returns to already-scarce capabilities (skill-augmenting AI).
- Practical priorities:
  - Diagnose and lower implementation costs where productive capability is weakest: data, management, cognitive skills, infrastructure, competition, finance, implementation capacity.
  - Prioritize task-specific tools that relax binding local constraints and embed in actionable workflows enabling human validation and institutional follow-through.
  - Do not treat exposure or technology acquisition as a productivity assumption in surveillance; scenario design should specify which margin AI is assumed to change (frontier productivity, returns to existing skills, or transition probabilities).
- Recognize gradualism: small increases in upward-movement probabilities each five-year step can compound into large multigenerational effects; policy should aim to shift transition probabilities upward at each step, especially in lower-tail economies.

### X. Limits and Caveats
- Markov matrices are descriptive summaries of historical transitions; they do not identify causal transition mechanisms.
- Decompositions are accounting diagnostics, not structural allocations.
- AI counterfactuals presented are conditional comparative-statics, not forecasts; the framework abstracts from labor displacement, capital ownership, changes in factor shares, and cross-border rents.
- The historical relationship between measured human capital and mobility does not imply that additional schooling will causally raise transition probabilities under all circumstances.

*International Monetary Fund — content unit wpiea2026190*

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

### References

### I. Introduction — Intelligence Divide and Human Capital
- Central research question: Will artificial intelligence (AI) help poorer countries catch up, or widen existing gaps by lowering the cost of using knowledge while favoring economies with existing skilled workers, reliable data, capable firms, and effective institutions?
- Key conceptual distinction:
  - Intelligence Divide: unequal capacity to convert increasingly cheap cognitive services into sustained productivity gains (not merely unequal access).
  - Empirical counterpart: productivity mobility — whether economies leave low-productivity states and move toward the frontier.
- Theoretical framing links:
  - Appropriate technology, absorptive capacity, and technology diffusion literatures (Atkinson and Stiglitz 1969; Lall 1992; Acemoglu and Zilibotti 2001).
  - Development accounting vs. diffusion/absorptive-capacity perspectives (Nelson and Phelps 1966; Benhabib and Spiegel 1994, 2005; Howitt and Mayer-Foulkes 2005).
- Operational research design:
  - Finite-state Markov framework in the tradition of Quah (1993, 1996) and others to study transition probabilities across relative-productivity states.
  - Caselli–Coleman decomposition to separate level effects (workforce composition) from mobility effects (probability of moving toward frontier).
- Core empirical pattern (from 1,097 country-period transitions, fixed 1975 non-U.S. quartiles):
  - Mean time required to leave the lowest state:
    - capital intensity: 16.1 years
    - measured human capital: 31.5 years
    - TFP: 44.0 years
  - Interpretation: observable inputs (capital, measured human capital) accumulate faster than they are translated into sustained relative-productivity improvements.
- Human capital and mobility:
  - Fixed within-year median split:
    - mean exit time from the lowest-productivity state for observations below the median: 79.1 years
    - mean exit time for observations above the median: 28.0 years
  - Continuous specifications with measured human capital improve prediction of subsequent transitions out of sample.
  - Evidence favors a capability gradient (systematic mobility contrast) rather than a single structural threshold.

### II. Background — Level vs. Mobility Channels
- Development accounting:
  - Decomposes output per worker into physical capital, measured human capital, and TFP.
  - Measured human capital explains part of cross-country income differences, but most of the gap remains in productivity (Hall and Jones 1999; Caselli 2005; Caselli and Ciccone 2013).
- Two channels for human capital:
  - Static/level channel: increases effective labor via larger skilled share or higher productivity premium for skilled labor; skill-biased AI enters here by raising A_H and thus depending on existing skilled supply and production systems.
  - Dynamic/absorptive-capacity channel: human capital accelerates adoption and implementation of frontier technologies (Nelson and Phelps 1966); success also depends on management, institutions, technology mismatch, credit constraints (Acemoglu, Akcigit, and Johnson 2026).
- Historical and sectoral examples:
  - Tractorization in sub-Saharan Africa: imported machinery often idle or uneconomic without credit, fuel, repair facilities, and complementary infrastructure (Binswanger and Donovan 1986).
  - Green Revolution: high-yielding seeds required local learning; imperfect knowledge slowed adoption and raised the return to schooling (Foster and Rosenzweig 1995, 1996).
  - Mobile phones in Kerala: rapid productivity gain where technology fit a concrete information bottleneck (Jensen 2007).
- Policy implication from background:
  - Measured schooling is an observable component of a wider capability bundle (education quality, management and organizational capacity, infrastructure, finance, state capability, institutions).
  - For AI to promote convergence, it must lower the costs of acquiring, interpreting, codifying, adapting, validating, and implementing knowledge where complementary capabilities are limited.

### III. Methods — Distribution Dynamics and Measurement
- Reduced-form empirical strategy:
  - Separates a level margin (Caselli–Coleman decomposition) from a mobility margin (Markov transition framework).
  - Purpose: characterize transition process governing relative productivity and isolate the margin on which AI would need to operate to affect relative development.
  - Results are descriptive and conditional counterfactuals, not causal estimates.
- Markov framework details:
  - Discrete-time stochastic process {X_t} with finite state space S={1,...,K}; transition matrix P = [p_ij] with p_ij = Pr(X_{t+τ}=j | X_t=i).
  - First-order Markov assumption used as a parsimonious approximation summarizing mobility conditional on current state.
  - Variables expressed relative to the United States: x̃ = x / x_{US,t}; United States used as a stable benchmark frontier.
  - Transitions measured over five-year intervals.
- Summary statistics and long-run measures:
  - Reported objects: transition probabilities, mean exit times, mean first-passage times, and a convergence statistic based on the second-largest eigenvalue of P.
  - Exit times: expected time required to leave a state; first-passage times: expected time to reach another state for the first time.
  - Convergence statistic (Kremer et al. 2001):
    - γ = − log_2 ( log |λ_2| )
    - λ_2 is the second-largest eigenvalue of P in absolute value.
    - γ is measured in transition periods; when reporting years, multiply γ by the interval length. Lower γ corresponds to faster convergence toward the stationary distribution.
  - Stationary distribution ψ satisfies ψ = ψP; used as a conditional long-run summary (not a forecast).
- Discretization and robustness:
  - Baseline uses fixed thresholds so moving upward closes a fixed part of the frontier gap.
  - Robustness checks: absolute thresholds, time-varying quartiles, terciles, quintiles, and a longer transition horizon.
  - Transition probabilities are sample-frequency maximum-likelihood estimates; standard errors follow Anderson and Goodman (1957), with country-block bootstrap inference where appropriate.

### Key empirical findings and policy-relevant conclusions
- Empirical findings:
  - Observed asymmetry: capital intensity and measured human capital move upward more readily than TFP across the same sample transitions.
  - Measured human capital is a modest explanatory factor in cross-country productivity levels but a meaningful marker of productivity mobility.
  - Economies with higher measured human capital face systematically greater mobility out of low-productivity states; the mobility contrast persists across fixed-rank, continuous, residualized, and out-of-sample exercises.
- Policy-relevant conclusions:
  - Access to AI alone does not equate to capacity to benefit; the relevant policy question is whether AI relaxes capability constraints that limit productivity mobility.
  - Expanding education, capital, digital infrastructure, or AI may be necessary but not sufficient for convergence; complementary investments in education quality, management, organizational capacity, infrastructure, finance, and state capability matter for converting inputs into sustained productivity improvements.
  - For AI to promote convergence, it must change the process by which economies acquire, adapt, and use productive knowledge (diffusion-enhancing AI), not merely raise returns to already-scarce capabilities (skill-augmenting AI).

*International Monetary Fund — content unit wpiea2026190*

### 3.2 Decomposing relative productivity

### 3.2 Decomposing relative productivity

### Methodological framework
- Output per worker in country i is specified as: Y_i = K_i^α (A_i E_i)^{1−α}, where K_i is physical capital, A_i total factor productivity, and E_i effective labor.
- The capital share α is set to one-third throughout.
- Effective labor is constructed from skilled and unskilled labor using a CES aggregator:
  - E_i = [ (A_{u,i} L_{u,i})^{(σ−1)/σ} + (A_{s,i} L_{s,i})^{(σ−1)/σ} ]^{σ/(σ−1)},
    where L_{u,i} and L_{s,i} denote unskilled and skilled labor, and σ is the elasticity of substitution between the two groups.
- The decomposition used is:
  - log(TFP_i / TFP_US) = log(A_{u,i} / A_{u,US}) + log(B_i / B_US) + ε_i,
    where B is the effective-labor term implied by workforce composition under the maintained common skill-premium assumptions, and the first term is a relative-efficiency component.

### Interpretation of decomposition
- The first term (relative-efficiency component) should not be interpreted as a pure technology measure; it captures relative efficiency and organizational margins.
- The residual ε_i collects what the two-part accounting representation does not explain.
- The exercise asks how much of the observed productivity gap workforce composition can account for before broader efficiency, organization, and implementation margins must enter.
- A small composition term implies the direct static channel of human capital is insufficient; human capital may matter through transition processes (absorbing, adapting, implementing technologies).
- The decomposition is a diagnostic bridge to mobility analysis, not a structural allocation of TFP to named mechanisms.

### Key assumptions and robustness checks
- Baseline imposes a common skill premium across countries (following Caselli and Coleman (2006)).
- Alternative elasticities of substitution between skilled and unskilled labor are reported, including:
  - the preferred Caselli–Coleman calibration,
  - a Cobb-Douglas benchmark,
  - higher-substitution cases.
- Robustness question: whether any plausible calibration makes workforce composition large enough to overturn the mobility interpretation.

### Relationship to mobility analysis
- The decomposition is deliberately narrow: it measures the direct static contribution of workforce composition to relative productivity; it does not attribute transition dynamics or broader capability formation to human capital directly.
- Results inform whether workforce composition can explain observed TFP gaps or whether mobility/efficiency factors must be invoked.

### 3.3 Human capital and mobility (method summary)
- Transition matrices are estimated separately for human-capital regimes; the data select the split:
  - For each candidate threshold τ, observations below and above τ are assigned separate transition matrices.
  - The likelihood gain relative to a pooled matrix is recorded.
  - Candidate cutoffs must leave adequate observations in each regime.
  - The selected threshold maximizes the likelihood-ratio statistic.
- The exercise tests for heterogeneous mobility regimes; it is not an attempt to impose convergence clubs by construction.
- The threshold is a diagnostic ("capability regime") indicating where the transition process changes sharply; it is not a causal treatment effect or mechanical tipping point.
- Complementary check: regression-adjusted mobility specifications of the form:
  - Pr(s_{i,t+τ} > s) = F(α_s(i,t) + β h + Γ Z + ρ_r + δ_t),
    where s_{i,t} is productivity state, h_{i,t} measured human capital, Z_{i,t} controls including log income per capita, and ρ_r and δ_t are region and period fixed effects.
- These regressions are supporting checks; they do not identify causal effects of schooling.

### 3.4 AI counterfactuals (comparative-static exercises)
- Two interpretations:
  1. AI as skill-augmenting technological change:
     - A_{s,i}' = (1 + θ) A_{s,i}, θ > 0.
     - Under this channel, gains depend on existing supply of skilled labor and the production systems where it is used; economies with larger skilled workforces benefit more immediately.
     - Captures possibility that AI raises return to existing capabilities without changing capability-acquisition processes.
  2. AI as a diffusion technology:
     - Lowers the cost of codifying, translating, diagnosing, adapting, and applying knowledge, raising the probability that low-productivity economies move upward.
     - This channel operates through changes in the transition matrix rather than the workforce-composition term.
- Neither exercise is a forecast; they illustrate what would have to be true for AI to narrow rather than reinforce relative development gaps.
- Framework abstracts from labor displacement, capital ownership, changes in factor shares, and cross-border rents.

### Data sources and construction
- Primary sources combined:
  - Maddison Project: long-run evolution of relative income.
  - Penn World Table 11.0 (PWT 11.0): measures of output, capital, employment, human capital, and TFP; used for production-function and mobility exercises over 1960–2015 in non-overlapping five-year intervals.
  - Barro-Lee educational-attainment data (updated through 2015): skilled and unskilled labor shares for the Caselli–Coleman decomposition.
- Specifics:
  - Main production sample: PWT 11.0 over 1960–2015 in non-overlapping five-year intervals.
  - A separate 2020 PWT cross-section is used only to illustrate the accounting decomposition and does not enter baseline mobility estimates.
  - Skilled labor: share of population aged 25+ with completed secondary education or more (Barro and Lee).
  - An auxiliary mobility comparison uses Hanushek and Woessmann’s (2012) cognitive-skill data.
- Variable treatments:
  - All principal variables are expressed relative to the United States.
  - Mobility analysis uses PWT human-capital index, rescaled so the United States equals one in each year.
  - Decomposition uses Barro-Lee attainment shares.
- Coverage and samples:
  - Pooled decomposition uses 120 countries and 1,186 country-period observations.
  - Baseline relative-TFP transition matrix uses 1,097 five-year transitions; capital-output and human-capital matrices use larger samples.
  - Each observed country-period transition receives equal weight.
  - Baseline classifies relative TFP into four states using thresholds from the 1975 cross-section, held fixed thereafter.
- Estimation and inference:
  - Transition probabilities estimated from observed frequencies.
  - Anderson-Goodman standard errors reported where appropriate.
  - Country-block bootstrap procedures used for threshold inference and nonlinear functions of estimated matrices; threshold exercise uses 300 bootstrap replications.
- Noted data limitations:
  - Educational attainment is an imperfect proxy for capability (does not capture school quality, cognitive skill, management, technological adaptability).
  - International productivity comparisons sensitive to PPP adjustments, national-accounts quality, and TFP construction, especially in lower-income economies.
  - The United States is a benchmark rather than a universal sectoral frontier.
  - Sample retains economies with complex transition histories (including successor states of former Soviet Union and Yugoslavia) when observations are usable.

### Key empirical findings on relative persistence
- Long-run benchmark: countries’ positions in the relative income distribution are highly persistent even when absolute incomes rise.
- Persistence in five-year transitions (Maddison Project data, 1960–2015; states defined from 1975 cross-section and held fixed):
  - Economies in the lowest state remain there with probability 0.943 over five years.
  - Economies in the highest state remain there with probability 0.931 over five years.
- Mobility patterns:
  - Movements are mostly local; jumps from the bottom to the top are absent.
  - The lower tail is durable and not merely a cross-section artifact.
  - Convergence in averages can coexist with lower-tail immobility; distribution dynamics distinguish catch-up from relative decline at the top.

*Source: IMF staff, "3.2 Decomposing relative productivity" (extracted from the provided chapter).*

### Appendix A translates these probabilities into development time. The mean exit time from the lowest income

### wpiea2026190 - Appendix A translates these probabilities into development time. The mean exit time from the lowest income

### Development-time interpretation
- The mean exit time from the lowest income state is 87.7 years.
- The mean first-passage time from the bottom to the top is 330.8 years.
- These are not forecasts; they express persistence embedded in the estimated process in interpretable units.
- A five-year persistence probability of 0.943, when repeated as a statistical process, corresponds to multi-generational persistence.
- Transition probabilities average over episodes (including conflict and debt crises) and summarize the realized postwar transition process rather than a structural law that applies unchanged in crises and tranquil periods.

### Proximate determinants: capital, human capital, and productivity
- If capital, human capital, and TFP all move slowly, this indicates broad failure of accumulation; if inputs move more readily than TFP, the bottleneck is converting inputs into productivity.
- Transition framework applied to: capital-output ratio, measured human capital, and relative TFP.
- Clear asymmetry: capital and human capital display more directional mobility; relative TFP is more persistent, especially in the lower tail.
- The world economy has narrowed gaps in measurable inputs more readily than gaps in productive efficiency.

### Transition-pattern findings (Figure 2 summary)
- Capital-output ratios:
  - Five-year probability of leaving bottom state: 0.327.
  - Movement from middle states often toward higher states.
  - Capital deepening direction is clear though not frictionless.
- Human capital:
  - Bottom-state five-year upward-mobility probability: 0.156.
  - Downward movements are rare; top state is highly persistent.
  - Measured human capital behaves like an accumulating stock.
- Relative TFP:
  - Lowest-state five-year persistence: 0.886 (so upward movement from Q1 is 0.114).
  - From Q3, downward movement to Q2 is more common than upward movement to Q4.
  - Implied stationary mass in Q4: 8.9 percent (compared with 77.1 percent for capital-output and 90.3 percent for human capital).

### Mobility statistics and comparisons (Table 2)
- Capital-output ratio (K/Y):
  - P(Q1,Q1): 0.673
  - Mean exit Q1 (years): 15.3
  - P(Q1→Q2,Q3,Q4): 0.327
  - P(Q4,Q4): 0.921
  - Stationary Q4 mass: 0.771
  - Number of transitions (NT): 1208
- Human capital (PWT hc):
  - P(Q1,Q1): 0.844
  - Mean exit Q1 (years): 32.1
  - P(Q1→Q2,Q3,Q4): 0.156
  - P(Q4,Q4): 0.988
  - Stationary Q4 mass: 0.903
  - Number of transitions (NT): 1441
- Relative TFP (PWT ctfp):
  - P(Q1,Q1): 0.886
  - Mean exit Q1 (years): 44.0
  - P(Q1→Q2,Q3,Q4): 0.114
  - P(Q4,Q4): 0.660
  - Stationary Q4 mass: 0.089
  - Number of transitions (NT): 1097
- Stricter comparison on same 1,097 transitions and 1975 non-U.S. quartiles:
  - Mean exit from Q1: capital intensity 16.1 years; human capital 31.5 years; TFP 44.0 years.
  - Upward mobility: 0.310 (capital), 0.159 (human capital), 0.114 (TFP).
- Interpretation:
  - Observable inputs tend to move upward; productivity mobility is local, reversible, and unlikely to carry countries into the near-frontier state.
  - As gaps in capital and schooling narrow, productivity dynamics become increasingly important for relative income.

### Relative TFP transition matrix (Table 3, 1960–2015)
- Five-year transition probabilities (From / To: <0.65, <0.88, <1.04, >=1.04) with N_transitions per row:
  - From <0.65 (N_transitions 449):
    - 0.886, 0.111, 0.002, 0.000
    - Standard errors: (0.015) (0.015) (0.002) (0.000)
  - From <0.88 (N_transitions 297):
    - 0.185, 0.640, 0.155, 0.020
    - Standard errors: (0.023) (0.028) (0.021) (0.008)
  - From <1.04 (N_transitions 207):
    - 0.019, 0.271, 0.551, 0.159
    - Standard errors: (0.010) (0.031) (0.035) (0.025)
  - From >=1.04 (N_transitions 144):
    - 0.000, 0.056, 0.285, 0.660
    - Standard errors: (0.000) (0.019) (0.038) (0.039)
- Stationary distribution: 0.479, 0.278, 0.154, 0.089 (for the four TFP states)
- Key patterns:
  - Lower tail is sticky; transitions out of Q1 are overwhelmingly local; jumps from the bottom to the frontier are absent.
  - Middle is not a smooth ladder; productivity mobility is constrained rather than a common escalator.

### Decomposing relative productivity: Caselli–Coleman decomposition (Figure 3 and Table 4)
- Decomposition separates workforce-composition term from relative-efficiency term and a residual; not structural.
- Visual finding: cross-country variation is much greater along the relative-efficiency axis than the workforce-composition axis.
- Conclusion: direct workforce-composition term is too small to account for most observed productivity differences; human capital likely matters dynamically by supporting absorption, adaptation, and implementation of technologies.
- Table 4 (Auxiliary 2020 cross-section, Caselli–Coleman decomposition):
  - Q1 (low HC), N 29:
    - Mean rel. TFP obs.: 0.486
    - log(TFP/TFP_US): -0.782
    - log(A_u / A_u_US): -2.469
    - log(B / B_US): 0.312
    - log residual: 1.375
  - Q2, N 29:
    - Mean rel. TFP obs.: 0.657
    - log(TFP/TFP_US): -0.484
    - log(A_u / A_u_US): -1.52
    - log(B / B_US): 0.371
    - log residual: 0.665
  - Q3, N 28:
    - Mean rel. TFP obs.: 0.727
    - log(TFP/TFP_US): -0.368
    - log(A_u / A_u_US): -0.991
    - log(B / B_US): 0.23
    - log residual: 0.393
  - Q4 (high HC), N 29:
    - Mean rel. TFP obs.: 0.759
    - log(TFP/TFP_US): -0.304
    - log(A_u / A_u_US): -0.671
    - log(B / B_US): 0.079
    - log residual: 0.288
- Robust conclusion: workforce composition contributes but does not dominate productivity differences; residual is largest in the lowest human-capital quartile.

### Human capital and productivity mobility: an estimated capability threshold
- Motivation: stronger measured human capital may be associated with faster exit from low-productivity states via complementarities (Nelson–Phelps intuition).
- Method: endogenous sample split (Hansen-style) selecting cutoff tau to maximize likelihood-ratio; candidate cutoffs must leave adequate cell sizes.
- Estimated threshold and inference (Table 5):
  - Endogenous HC threshold τ̂: 0.676
  - 95% bootstrap CI lower: 0.398
  - 95% bootstrap CI upper: 0.841
  - LR at τ̂: 60.57
  - Bootstrap replications: 300
  - Mean exit time from Q1 | HC < τ̂ (years): 64.8
  - Mean exit time from Q1 | HC ≥ τ̂ (years): 25.4
  - Exit-time ratio (Low/High): 2.55×
  - N transitions, HC < τ̂: 632
  - N transitions, HC ≥ τ̂: 476
- Operational cutoff used for comparisons: 0.64 (chosen for comparability; close to point estimate and identical to upper human-capital state boundary used elsewhere).
- Regime contrasts (narrative and Table 6 summary):
  - Low-capability regime: Q1 persistence 0.922; upward movement over five years 7.8 percent; stationary distribution assigns 68.8 percent mass to Q1 and 5.2 percent to Q4.
  - High-capability regime (above operational cutoff): Q1 persistence 0.826; upward mobility 17.4 percent; stationary Q4 mass 12.8 percent.
  - Under operational split, mean exit time from Q1 falls from 64.3 to 28.7 years; expected passage from Q1 to Q4 falls from 368.5 to 177.0 years.
- Cross-sectional check (Figure 5):
  - In 2015 cross-section, no country combines lowest human-capital state with frontier-level TFP; pooled panel such observations are rare.
  - Asymmetry: schooling without frontier productivity is common; sustained frontier productivity with very low measured human capital is difficult to find.

### Regression-adjusted mobility and interpretation (Table 7 summary)
- Discrete specifications show regime contrast concentrated among economies that begin in the lowest-productivity state, Q1.
- When controlling flexibly for exact initial productivity, the coefficient on continuous human capital falls:
  - From 0.417 to 0.150 (p = 0.068) when adding exact initial productivity control.
  - Adding initial income reduces coefficient further to 0.028 (p = 0.790).
- Interpretation caveats:
  - The threshold result should not be read as an independent causal effect of human capital or evidence of a sharp structural tipping point.
  - A continuous mobility measure (five-year growth in log relative TFP) yields a more robust pattern: conditional on a cubic in exact initial productivity, a one-unit increase in human capital relative to the United States is associated with higher TFP growth (estimates reported in the main text beyond supplied excerpt).

*Source: Authors’ calculations from Penn World Table 11.0 (content excerpt from wpiea2026190).*

### 0.222 higher five-year growth in log relative TFP. The coefficient remains 0.200 after controlling additionally for

### wpiea2026190 - 0.222 higher five-year growth in log relative TFP. The coefficient remains 0.200 after controlling additionally for

### Human capital, productivity mobility, and interpretation
- Estimated association: "0.222 higher five-year growth in log relative TFP." The coefficient remains "0.200" after controlling additionally for income and capital intensity (p < 0.001).
- Similar estimates are obtained for: welfare-relevant TFP, national-price TFP growth, and labor-productivity growth.
- Country fixed effects: weaken precision of the ctfp estimate, although the coefficient remains positive; national-price TFP estimate remains positive and statistically significant.
- Nature of regressions: descriptive rather than causal — they show that the human-capital relationship is not confined to crossing an arbitrarily defined state boundary.
- Median split mobility evidence:
  - Mean exit time from Q1 for observations below the within-year median: "79.1 years".
  - Mean exit time from Q1 for observations above the within-year median: "28.0 years".
  - Equality of full transition matrices rejected in a 1,000-replication country-block permutation test (p = 0.001).
- Predictive evidence:
  - Fixed-rank specification improves both early-to-late and late-to-early prediction and performs better in eight of ten country folds.
  - When human capital enters smoothly rather than through a discrete split, it improves held-out transition prediction in all fifty repeated country-fold assignments.
  - Human capital residualized on income, period effects, and initial productivity state continues to contain information about the transition matrix (p = 0.022).
- Conceptual distinction: measured human capital contributes modestly to static productivity differences but is informative about expected duration of low-productivity states and subsequent productivity growth — difference between a stock (schooling increases effective labor) and a conversion process (capability bundle influences conversion of skills, capital, and knowledge into reliable production).

### AI: skill augmentation versus diffusion
- Two distinct channels:
  - Skill augmentation (skill-biased technological change): raises productivity of skilled labor; identical access can generate different aggregate gains depending on skill distribution and composition effects.
  - Diffusion (lowering costs of learning, validation, adaptation, and implementation): changes the transition process and can promote convergence if it raises upward mobility in low-productivity economies.
- Key conceptual points:
  - Technical capability vs productive capability: AI system capability does not imply firms/institutions can use it reliably at scale.
  - Exposure ≠ adoption; adoption ≠ economic transformation.
  - Preparedness (observable conditions for adoption) is distinct from historical mobility (capability regimes outcome-based classification).
- Complementarity requirements for diffusion to raise mobility: data, management, cognitive skills, infrastructure, competition, finance, implementation capacity.
- Empirical complement: Che, Xin, and Yoshida (2026) — better-prepared economies adopt AI earlier; reforms strengthening fundamentals bring adoption forward.
- For follower economies, adaptation (fitting tools to local data, workflows, constraints, institutions) is central; adoption alone is insufficient.

### Accounting sensitivity exercise (Table 8) — AI as a skill-biased premium shock
- Purpose: trace workforce-composition term under stagnant and rising human-capital paths across AI premium calibrations; not an estimate of aggregate AI effects; no probabilities attached.
- Table 8 entries (AI scenario / HC path / log B (High HC) / log B (Low HC) / Composition gap ∆B):
  - Baseline (π =1.80) Stagnant HC: "1.476" | "1.563" | "-0.088"
  - Baseline (π =1.80) Rising HC: "1.476" | "1.653" | "-0.177"
  - Moderate AI (π =2.50) Stagnant HC: "1.706" | "1.732" | "-0.025"
  - Moderate AI (π =2.50) Rising HC: "1.706" | "1.85"  | "-0.143"
  - Strong AI (π =3.00) Stagnant HC: "1.838" | "1.829" | "0.009"
  - Strong AI (π =3.00) Rising HC: "1.838" | "1.963" | "-0.125"
- Interpretation: as the AI premium rises, the composition gap narrows and can change sign under stagnant human capital; continued skill accumulation offsets part of the premium effect. Skill-biased AI does not mechanically cause convergence or divergence — outcome depends on initial skill distribution and subsequent capability formation.

### Diffusion channel and practical implementation
- Diffusion can reduce costs of codifying tacit knowledge, translating technical information, diagnosing production problems, and adapting frontier practices.
- Aggregate productivity gains from diffusion may be delayed or initially negative during installation due to required accumulation of data, training, workflow redesign, and organizational capital (Brynjolfsson, Rock, and Syverson 2021).
- For follower economies, adaptation is pivotal: need fit to local languages, data, regulations, production processes, service-delivery systems.
- Electrification parallel: large gains required organizational reorganization, not only equipment replacement (David 1990).
- The most valuable application may be task-specific tools that relax binding local constraints and embed in actionable workflows.
- Sub-Saharan Africa: immediate challenge is ability to adopt, adapt, and scale AI amid constraints in electricity, digital infrastructure, technical skills, regulatory and institutional capacity; these constraints map closely to the capability bundle.

### Long-run projections under alternative mobility regimes (Table 9 and Figure 7)
- Scenarios considered: Baseline (unconditional relative-TFP matrix), High-HC transition matrix, Broad AI diffusion (scale =1.0), Unequal AI diffusion (scale =1.0). Scenarios are conditional illustrations, not forecasts.
- Key comparative outcomes:
  - Applying the high-capability transition matrix substantially shortens persistence:
    - After 100 years: Q1 share = "0.262" rather than "0.482"; Q4 share = "0.124" rather than "0.088".
    - Mean exit time from Q1 falls from "44.0" to "28.7" years.
    - Passage time Q1→Q4 falls from "241.8" to "177.0" years.
    - Interpretation: materially shorter duration in lower tail, not rapid convergence.
  - Broad AI diffusion improves mobility without eliminating persistence:
    - After 100 years: Q1 = "0.340", Q4 = "0.105".
    - This regime has the fastest convergence statistic and a mean exit time similar to the high-capability case.
  - Unequal AI diffusion is the cautionary case:
    - After 100 years: Q1 = "0.545", stationary share = "0.543".
    - Mean exit time rises to "58.8" years; passage time to Q4 to "285.9" years.
    - Interpretation: AI is present but mobility benefits accrue where complementary capabilities already exist.
- Table 9 selected entries (Horizon / Q1 / Q2 / Q3 / Q4 / γ (years) / MET1 (years) / Passage Q1→Q4 (years)):
  - Baseline (unconditional TFP) 25 years: "0.510" | "0.276" | "0.140" | "0.074" | "21.2" | "44.0" | "241.8"
  - Baseline (unconditional TFP) 100 years: "0.482" | "0.277" | "0.153" | "0.088" | "21.2" | "44.0" | "241.8"
  - High-HC transition matrix 25 years: "0.356" | "0.360" | "0.206" | "0.077" | "20.6" | "28.7" | "177.0"
  - High-HC transition matrix 100 years: "0.262" | "0.353" | "0.262" | "0.124" | "20.6" | "28.7" | "177.0"
  - Broad AI diffusion (scale =1.0) 100 years: "0.340" | "0.355" | "0.200" | "0.105" | "17.2" | "28.6" | "187.6"
  - Unequal AI diffusion (scale =1.0) 100 years: "0.545" | "0.235" | "0.133" | "0.087" | "25.2" | "58.8" | "285.9"
- Mechanism emphasis: AI compresses the lower tail only if lower implementation costs raise upward mobility in low-productivity economies; widespread AI can coexist with stratified productivity mobility.

### Robustness, stability, and timing sensitivity (Tables 10–11)
- Transition horizon check (Table 10) — Chapman-Kolmogorov consistency:
  - Comparison: directly estimated ten-year matrix P₁₀ vs two-step five-year matrix P₅².
  - Statistics:
    - γ (years): 5-year baseline "21.186" vs 10-year direct "12.102" | Difference "-9.084"
    - MET Q1 (years): "44.020" vs "29.923" | Difference "-14.097"
    - Passage Q1→Q4 (years): "241.825" vs "149.076" | Difference "-92.749"
    - Stationary Q1: "0.479" vs "0.478" | Difference "-0.001"
    - Stationary Q4: "0.089" vs "0.085" | Difference "-0.004"
    - ‖P₁₀ − P₅²‖F: "0.098"
    - Max |entry|: "0.041"
    - N transitions (10-year): "978"
  - Interpretation: direct ten-year matrix implies faster movement than five-year specification, but almost the same long-run distribution; timing estimates are horizon-sensitive, but lower-tail conclusion persists.
- Temporal stability (Table 11) — pre- and post-1995 comparison:
  - Early period (<1995) vs Late period (≥1995):
    - Q1 persistence: "0.870" vs "0.911"
    - Upward mobility from Q1: "0.130" vs "0.089"
    - Mean exit time Q1 (years): "38.409" vs "56.190"
    - Passage time Q1→Q4 (years): "152.481" vs "452.799"
    - Stationary Q1 mass: "0.350" vs "0.612"
    - Stationary Q4 mass: "0.154" vs "0.036"
    - N transitions: "517" vs "476"
    - χ2 homogeneity test (p-value): "0.4500"
  - Interpretation: the chi-squared homogeneity test does not reject equality of the transition matrices at conventional levels; point estimates show, if anything, greater lower-tail persistence after 1995.

### Policy-relevant implications and actionable priorities
- To translate AI availability into convergence, focus on lowering implementation costs where productive capability is weakest: investments in data, management, cognitive skills, infrastructure, competition, finance, and implementation capacity.
- Pair adoption with investments that make adaptation productive — do not treat AI access alone as sufficient.
- Prioritize task-specific tools that relax binding local constraints and can be embedded in workflows enabling effective human validation and institutional follow-through.
- Recognize gradualism: small increases in upward-movement probabilities at each five-year step can compound into large effects over generations; policy should aim to shift transition probabilities upward at each step, especially in lower-tail economies.

*Source: wpiea2026190 (IMF working paper content provided).*

### 0.870 to 0.911 and upward mobility falls from 0.130 to 0.089. Wider access to global technology and markets

### wpiea2026190 - 0.870 to 0.911 and upward mobility falls from 0.130 to 0.089. Wider access to global technology and markets

### Key empirical findings on mobility and capability regimes
- Fixed 1975 quartiles (baseline) produce high persistence in the lowest productivity state: P(Q1,Q1) = 0.886 and upward Q1 = 0.114 (N = 1097).
- Under alternative discretizations mobility remains slow:
  - Absolute thresholds [0.20,0.40,0.70]: P(Q1,Q1) = 0.400, Upward Q1 = 0.600, Exit Q1 = 8.3 years, Passage B→T = 71.0 years, Stationary bottom = 0.009, Stationary top = 0.509 (N = 1097).
  - Time-varying quartiles: P(Q1,Q1) = 0.829, Upward Q1 = 0.171, Exit Q1 = 29.2 years, Passage B→T = 159.2 years, Stationary bottom = 0.264, Stationary top = 0.194 (N = 1097).
  - Terciles: P(Q1,Q1) = 0.863, Upward Q1 = 0.137, Exit Q1 = 36.5 years, Passage B→T = 103.4 years, Stationary bottom = 0.358, Stationary top = 0.293 (N = 1097).
  - Quintiles: P(Q1,Q1) = 0.820, Upward Q1 = 0.180, Exit Q1 = 27.8 years, Passage B→T = 205.9 years, Stationary bottom = 0.214, Stationary top = 0.154 (N = 1097).
- Post-1995 period (wider access to digital technologies and global value chains) did not make the lower tail noticeably more mobile: diffusion of tools can coexist with persistent differences in the capability to turn them into productivity.

### Robustness checks and alternative measures
- Discretization sensitivity:
  - Fixed 1975 quartiles preserve historical distribution meaning; absolute thresholds ask a different question (crossing fixed low levels) and show more apparent convergence.
  - Qualitative result (slow mobility of the lower tail) survives fixed quartiles, time-varying quartiles, terciles, and quintiles.
- Cognitive-skills robustness (Hanushek–Woessmann, 69-country sample, split at median HW score 4.69):
  - Low cognitive skills (HW score ≤ 4.69): P[Q1,Q1] = 0.881 (SE 0.028), Mean Exit Q1 = 41.9 years, Passage Q1→Q4 = 196.6 years, Stationary Q1 = 0.467, Stationary Q4 = 0.087, N transitions = 334, N countries = 34.
  - High cognitive skills (HW score > 4.69): P[Q1,Q1] = 0.818 (SE 0.047), Mean Exit Q1 = 27.5 years, Passage Q1→Q4 = 111.0 years, Stationary Q1 = 0.098, Stationary Q4 = 0.235, N transitions = 321, N countries = 35.
  - All countries (restricted sample): P[Q1,Q1] = 0.860 (SE 0.025), Mean Exit Q1 = 35.7 years, Passage Q1→Q4 = 147.3 years, Stationary Q1 = 0.279, Stationary Q4 = 0.150, N transitions = 655, N countries = 69.
- Appendix E: varying elasticity of substitution between skilled and unskilled labor changes workforce-composition term but does not dominate productivity gap under reported calibrations.
- Frontier and weighting checks:
  - Replacing U.S. frontier with the 90th percentile, 95th percentile, or geometric mean of the five leaders changes waiting times but preserves the low- versus high-capability ordering.
  - Population weighting lengthens lower-tail persistence mainly due to China and India; excluding them returns estimates close to unweighted results.
  - Leave-one-country-out threshold searches preserve higher-capability exit advantage in every case.

### Interpretation: capability vs. accumulation
- Distinction emphasized: observable accumulation (capital intensity, measured human capital) versus capability to convert inputs into sustained productivity gains (mobility).
- Mean times to leave the lowest quartile on identical sample and harmonized state construction:
  - Capital intensity: 16.1 years to leave lowest quartile.
  - Measured human capital: 31.5 years to leave lowest quartile.
  - TFP: 44.0 years to leave lowest quartile.
- Fixed within-year median split implies mean exit times from the lowest-productivity state of:
  - 79.1 years for countries below the median.
  - 28.0 years for countries above the median.
- Human capital is better read as an observable marker of an inherited capability bundle than as a sole causal threshold; controlling flexibly for initial productivity and income weakens a discrete exit coefficient, consistent with a capability-related mobility gradient rather than a universal tipping point.

### AI implications and policy recommendations
- Core insight: access to AI and other technologies is necessary but not sufficient for convergence; the relevant margin is whether AI lowers the costs of learning, validation, adaptation, and implementation sufficiently to change mobility processes for economies far from the frontier.
- Possible channels for AI without convergence:
  - Expanding the frontier without altering relative positions.
  - Augmenting returns to existing skills and reinforcing complementarities that may widen gaps.
- Policy should focus on bottlenecks to productive adaptation, pairing adoption with the capability complements needed to act on AI outputs:
  - Skills, reliable data and infrastructure, management, finance, competition, and institutions able to experiment and learn.
- Empirical and policy margins to treat separately: exposure, access, adoption, adaptation, intensive use, productivity, and relative mobility.
- For IMF surveillance and capacity development:
  - Do not translate technological exposure mechanically into productivity assumptions.
  - Scenario design should specify which margin AI is assumed to change: frontier productivity, return to existing skills, or probability that low-productivity economies move upward.
  - Capacity development should diagnose constraints preventing productive use rather than treating technology acquisition as the objective.

### Limits and caveats
- Markov matrices characterize transitions but do not identify causal transition mechanisms; decompositions are accounting rather than structural.
- AI exercises presented are conditional comparisons, not forecasts.
- The results do not establish that additional schooling causally raises transition probabilities or that historical human-capital relationships will remain unchanged under AI.
- Other important margins outside the productivity-mobility question: labor displacement, capital ownership, factor shares, trade, and international allocation of AI rents.

*Source: Authors’ calculations from Penn World Table 11.0.*

### Appendix A: Relative Income Dynamics

### Appendix A: Relative Income Dynamics

### Mobility statistics and definitions
- Mobility statistics are implied by the relative-income transition matrix in Table 1.
- Exit and passage times are reported in years.
- The convergence statistic is defined in Section III.
- Source of calculations: Authors’ calculations from the Maddison Project Database 2020.

### Mean first-passage and recurrence times
- Off-diagonal cells report mean first-passage times from the row state to the column state, computed from the preceding income transition matrix.
- The MFR row reports mean first-recurrence times, 1/pi_i, in years.
- The implied Q1-to-Q4 passage time is 330.8 years.
- Source of calculations: Authors’ calculations from the Maddison Project Database 2020.

*Source: Authors’ calculations from the Maddison Project Database 2020.*

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_Source: https://www.imf.org/-/media/files/publications/wp/2026/english/wpiea2026190.pdf_