## Annex I. Quintiles and mobility

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### Unique Stationary Distributions
- Methodology:
  - Use finite state Markov chains to study mobility, drawing on ideas in Quah (1993), Feyrer (2008), and Im and Rosenblatt (2015).
  - Employ transition probability matrices; when the transition probability from middle-income to high-income is low compared to other upward transitions, that suggests greater difficulty achieving high-income status (a “middle-income trap” interpretation).
  - Complement transition analysis with mean passage times derived from transition matrices to capture the influence of both upwards and downwards mobility.
  - Systematic use of “constant growth thresholds” to define income categories; argued to be the default choice for middle-income trap analysis.
- Empirical scope:
  - Long-span GDP data from 1950 to 2015 (65 years), using five-year intervals, yielding more than 1,800 transitions between income categories.
  - Sample represents more than 93 percent of the world population in 2018.
  - Additional analysis using Penn World Table (PWT) version 10, covering 1975–2015.
- Key findings:
  - Mobility within the distribution of output per head is low.
  - There is no long-run tendency for absolute convergence in output per head.
  - Little evidence that transitions from a middle-income category are uniquely difficult when analyzing output dynamics alone.

### Mean Passage Times
- Rationale and metric:
  - Mean passage times (MPT) provide the average time to move between particular states, conditioning on both upward and downward moves.
  - Introduce a continuous summary statistic ∆MPT: the difference in mean passage times between the upwards transition to the highest category and the upwards transition to the next-highest category, to quantify trap extent.
- Empirical implications:
  - For GDP per head transitions (1950–2015 Maddison and 1975–2015 PWT), mean passage times and transition matrices show low mobility and no strong evidence of uniquely long middle-income persistence in output per head.
  - For proximate determinants:
    - Relative capital intensity and human capital show upwards mobility and convergence.
    - Relative TFP shows a lack of convergence; divergence in TFP has been masked by convergence in capital intensity and human capital.
    - If current patterns persist, future outcomes for output will be driven mainly by the TFP process.
- Policy-relevant interpretation:
  - A trap is operationalized when emerging from middle income is lengthy and uncertain compared with historical precedents; mean passage times permit continuous comparisons of protracted escape.

### Data, thresholds, and samples used
- Primary data sources:
  - Maddison Project Database 2020: 1950–2015; balanced panel of 145 countries observed at 14 five-year points → NT = 145×(14−1) = 1,885 transitions.
  - Penn World Table (PWT) version 10.01: typically 1975–2015; real GDP per adult of working age (15-64) used.
- Coverage:
  - Largest sample represents more than 7.1 billion people in 2018, or more than 93 percent of that year’s total population.
  - PWT sample represents 92.7 percent of 2018 global population (134 countries, T = 8, NT = 1,072 transitions).
- Primary constant growth thresholds relative to the U.S.: (0.09, 0.18, 0.36, 0.72).
  - Middle-income often defined as between 36% and 72% of US GDP per head.
- Thresholds for proximate determinants (K/Y, human capital h, TFP A): (0.32, 0.48, 0.72) (geometric factor 3/2).

### Aggregate GDP per Head — Key results (Maddison and PWT)
- Maddison (1950–2015; 145 countries; NT = 1,885):
  - Upwards mobility is slow and there are meaningful downwards transitions.
  - Stationary distribution places greatest mass on the highest output category (>72% of U.S.), yet countries will spend about two-thirds of their time below this category (no absolute convergence).
  - Evidence points to limited mobility rather than a sharply distinct middle-income trap in transition probabilities alone.
- PWT (1975–2015; 134 countries; NT = 1,072; T = 8):
  - Transition probabilities (five-year, selected entries with asymptotic std. errors):
    - From <0.09 to <0.09: 0.909 (0.015)
    - From <0.09 to <0.18: 0.086 (0.015)
    - From <0.18 to <0.09: 0.167 (0.026)
    - From <0.18 to <0.18: 0.685 (0.033)
    - From <0.36 to <0.36: 0.770 (0.030)
    - From <0.72 to <0.36: 0.777 (0.031)
    - From <∞ to <0.36: 0.889 (0.027)
  - Transition counts (rows): 
    - Row <0.09: 318, 30, 2, 0, 0
    - Row <0.18: 34, 139, 30, 0, 0
    - Row <0.36: 1, 23, 154, 22, 0
    - Row <0.72: 0, 2, 16, 143, 23
    - Row <∞: 1, 14, 120, 0, 0
  - Convergence statistic: γ = 82.6 (five-year periods).
  - Marginal distributions:
    - Last period: 0.276, 0.172, 0.209, 0.164, 0.179
    - +25 years: 0.289, 0.158, 0.190, 0.177, 0.186
    - +100 years: 0.290, 0.151, 0.179, 0.181, 0.199
    - Stationary ψ∗: 0.280, 0.148, 0.179, 0.185, 0.208
  - Mean passage times (PWT, years, rounded to one decimal place):
    - From <0.09 to <0.18: 62.5; to <0.36: 138.6; to <0.72: 283.9; to <∞: 445.4
    - From <0.18 to <0.09: 133.0; to <0.36: 89.5; to <0.72: 234.8; to <∞: 396.3
    - From <0.36 to <0.09: 249.9; to <0.18: 124.3; to <0.72: 145.3; to <∞: 306.8
    - From <0.72 to <0.09: 338.0; to <0.18: 211.7; to <0.36: 109.2; to <∞: 161.5
    - From <∞ to <0.09: 377.1; to <0.18: 250.9; to <0.36: 147.0; to <0.72: 54.7
    - Mean first recurrence times (MFR) by state: 17.8, 33.8, 28.0, 27.0, 24.0
  - Interpretation:
    - Stationary distribution for output per head is relatively flat; more than half the world’s countries are below the highest category in the stationary distribution.
    - Little evidence for a pronounced middle-income trap in transition probabilities alone; overall limited mobility and slow convergence dominate.

### Proximate determinants — approach and thresholds
- Framework: Cobb-Douglas Y = A K^β (hL)^{1−β}, focus on output per working-age adult and K/Y, h, and A distributions and transitions.
- Thresholds for proximate determinants: (0.32, 0.48, 0.72) (geometric factor 3/2).
- High-level finding:
  - Capital intensity (K/Y) and human capital converge upwards across countries.
  - Relative TFP shows much slower or absent convergence; lack of TFP convergence likely will dominate future relative output dynamics.

### Capital intensity (K/Y) — results (1975–2015)
- Five-year transition matrix (probabilities and standard errors):
  - Row <0.32: 0.660 (0.037), 0.241 (0.034), 0.093 (0.023), 0.006 (0.006)
  - Row <0.48: 0.047 (0.015), 0.524 (0.036), 0.340 (0.034), 0.089 (0.021)
  - Row <0.72: 0.004 (0.004), 0.101 (0.018), 0.552 (0.030), 0.343 (0.029)
  - Row <∞ : 0.005 (0.003), 0.049 (0.009), 0.946 (0.009)
- Convergence parameter: γ = 12.6.
- Distributional snapshots:
  - Last period: 0.000, 0.058, 0.129, 0.813
  - +25 years: 0.006, 0.039, 0.124, 0.831
  - +100 years: 0.007, 0.038, 0.122, 0.834
  - Stationary ψ*: 0.007, 0.038, 0.122, 0.834
- Transition counts and sample: NT = 1240, N = 155, T = 8.
- Mean Passage Times (years, rounded to one decimal place):
  - From <0.32: to <0.48: 95.3; to <0.72: 39.3; to <∞: 38.5
  - From <0.48: to <0.32: 2177.9; to <0.72: 32.2; to <∞: 26.8
  - From <0.72: to <0.32: 2390.1; to <0.48: 272.6; to <∞: 17.5
  - From <∞ : to <0.32: 2463.2; to <0.48: 340.3; to <0.72: 95.4
  - MFR row: 765.8, 132.6, 41.1, 6.0
- Interpretation:
  - Substantial upwards mobility in capital intensity and limited downwards mobility.
  - Distribution by 2010–2015 already close to stationary; unlikely that equilibrium-based middle-income trap is explained by capital intensity differences.

### Human capital — results (1975–2015)
- Human capital index: exp(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 s > 8
- Five-year transition matrix (probabilities and standard errors):
  - Row <0.32: 0.774 (0.075), 0.226 (0.075)
  - Row <0.48: 0.007 (0.005), 0.856 (0.020), 0.137 (0.020)
  - Row <0.72: 0.010 (0.005), 0.893 (0.015), 0.097 (0.015)
  - Row <∞ : 0.014 (0.007), 0.986 (0.007)
- Convergence parameter: γ = 36.2.
- Distributional snapshots:
  - Last period: 0.000, 0.186, 0.357, 0.457
  - +25 years: 0.003, 0.100, 0.309, 0.588
  - +100 years: 0.001, 0.026, 0.179, 0.794
  - Stationary ψ*: 0.000, 0.009, 0.122, 0.869
- Transition counts and sample: NT = 1032, N = 129, T = 8.
- Mean Passage Times (years, rounded to one decimal place):
  - From <0.32: to <0.48: 22.1; to <0.72: 59.6; to <∞: 114.9
  - From <0.48: to <0.32: 86536.9; to <0.72: 37.5; to <∞: 92.7
  - From <0.72: to <0.32: 90621.2; to <0.48: 4084.4; to <∞: 55.3
  - From <∞ : to <0.32: 90988.7; to <0.48: 4451.9; to <0.72: 367.5
  - MFR row: 19545.6, 565.7, 41.1, 5.8
- Interpretation:
  - Very little downwards mobility in relative human capital; some upwards mobility but slow into highest category.
  - Stationary distribution concentrates most mass in highest category; absolute convergence in human capital is expected eventually.
  - Even after another 25 years, about 40 percent of countries remain outside the highest category of relative human capital.

### Total Factor Productivity (TFP) — results (PWT ctpf, 1975–2015)
- Five-year transition matrix (probabilities and standard errors):
  - Row <0.32: 0.750 (0.082), 0.250 (0.082)
  - Row <0.48: 0.087 (0.029), 0.685 (0.048), 0.217 (0.043), 0.011 (0.011)
  - Row <0.72: 0.011 (0.008), 0.173 (0.028), 0.631 (0.036), 0.184 (0.029)
  - Row <∞ : 0.005 (0.003), 0.126 (0.016), 0.869 (0.016)
- Convergence parameter: γ = 23.8.
- Distributional snapshots:
  - Last period: 0.033, 0.200, 0.322, 0.444
  - +25 years: 0.073, 0.211, 0.278, 0.438
  - +100 years: 0.090, 0.227, 0.274, 0.409
  - Stationary ψ*: 0.092, 0.229, 0.274, 0.405
- Transition counts and sample: NT = 720, N = 90, T = 8.
- Mean Passage Times (years, rounded to one decimal place):
  - From <0.32: to <0.48: 20.0; to <0.72: 51.4; to <∞: 102.1
  - From <0.48: to <0.32: 197.4; to <0.72: 31.4; to <∞: 82.1
  - From <0.72: to <0.32: 248.9; to <0.48: 64.3; to <∞: 55.2
  - From <∞ : to <0.32: 285.3; to <0.48: 100.2; to <0.72: 39.4
  - MFR row: 54.3, 21.8, 18.3, 12.3
- Interpretation:
  - Upwards mobility exists but with noticeable downwards mobility, especially from the highest category.
  - Stationary distribution places significant mass on the second-highest category.
  - Even in the very long run, more than half the world’s countries will have TFP less than 72 percent of the U.S. level; sample limitation: 90 countries (~78 percent of 2018 world population).

### Common-sample and benchmarking results (basic TFP, N = 122)
- Common sample: NT = 976, N = 122, T = 8; basic TFP computed from equation (1) with β = 0.48 (Feenstra et al. average labor share 0.52 implying β = 0.48 under perfect competition).
- Statistic ∆MPT = absolute difference in mean passage times between upwards transition to highest category and to next-highest category.
- Findings, benchmarked to US (Table 11):
  - Output per w/a adult stationary: 0.619, 0.076, 0.107, 0.198; ∆MPT = -24.8
  - Basic TFP stationary: 0.224, 0.383, 0.255, 0.138; ∆MPT = 127.2
  - K/Y ratio stationary: 0.006, 0.040, 0.134, 0.820; ∆MPT = -11.2
  - Human capital stationary: 0.000, 0.006, 0.099, 0.894; ∆MPT = 17.2
  - Interpretation: Basic TFP shows very weak upwards mobility (∆MPT = 127.2 years), while capital intensity and human capital offset TFP stagnation for output per working-age adult.
- Findings, benchmarked to G7 median (Table 12):
  - Output per w/a adult stationary: 0.638, 0.104, 0.070, 0.188; ∆MPT = 81.4
  - Basic TFP stationary: 0.078, 0.208, 0.314, 0.401; ∆MPT = 29.6
  - K/Y ratio stationary: 0.029, 0.078, 0.216, 0.677; ∆MPT = 1.8
  - Human capital stationary: 0.000, 0.004, 0.131, 0.866; ∆MPT = 13.3
  - Interpretation: Benchmark choice (US vs G7 median) materially affects ∆MPT magnitudes and trap assessments.

### Alternative thresholds and sensitivity
- Alternative thresholds: <0.36, <0.48, <0.64, <∞ (Table 13; NT = 976; N = 122; T = 8).
  - Output per w/a adult stationary: 0.639, 0.058, 0.067, 0.236; ∆MPT = -9.7
  - Basic TFP stationary: 0.361, 0.287, 0.170, 0.182; ∆MPT = 71.7
  - K/Y ratio stationary: 0.014, 0.034, 0.088, 0.864; ∆MPT = -20.6
  - Human capital stationary: 0.000, 0.002, 0.031, 0.967; ∆MPT = 3.2
- Sensitivity note: using a lower β = 1/3 yields a mid-level trap for relative TFP as well, with even less upwards mobility into the top category.

### Robustness checks
- First-order Markov assumption and time aggregation (Maddison, 1950–2010): 10-year transitions compared with square of 5-year matrix — matrices “remarkably similar.”
  - Stationary (5-year) ψ*: 0.129, 0.095, 0.167, 0.237, 0.371
  - Stationary (10-year) ψ*: 0.096, 0.076, 0.141, 0.234, 0.453
- Excluding oil/mineral exporters (Maddison) produces little change in stationary distribution:
  - With vs Without oil/mineral producers:
    - <0.09: 0.130 vs 0.129
    - <0.18: 0.100 vs 0.090
    - <0.36: 0.174 vs 0.142
    - <0.72: 0.248 vs 0.265
    - <∞: 0.348 vs 0.374
- Subperiod stationary distributions (Maddison):
  - 1950–1980: 0.140, 0.101, 0.205, 0.224, 0.331
  - 1980–2015: 0.182, 0.145, 0.191, 0.230, 0.252
- Inference considerations:
  - Reporting transition counts gauges estimation precision.
  - Slow convergence in GDP per head implies stationary distributions sensitive to small changes in transition probabilities; proximate determinants with faster convergence yield more robust stationary distributions.
  - Benchmarks matter (US vs G7 median); choice of thresholds matters.

### Synthesis and implications
- Core synthesis:
  - Stationary distribution for output per head implies long-run dispersion—no absolute convergence.
  - Mean passage times generally argue against a distinct middle-income trap based solely on transition probabilities for output per head; rather, there is broad low mobility.
  - Proximate determinants diverge: capital-output ratios and human capital on course for absolute convergence; relative TFP lacks upwards mobility and may dominate long-run dynamics.
  - Consequence: over time, a middle-income trap for output per head may become more apparent once convergence of capital intensity and human capital no longer offsets lack of upwards mobility in relative TFP.
- Qualifications:
  - Results extrapolate from experience since mid-1970s; convergence processes may have changed in the most recent twenty years.
  - Higher absolute levels of human capital might enable future upwards mobility in relative TFP (to be investigated in Imam and Temple (2024)).

### Policy implications and research priorities
- Empirical approach: study middle-income trap using transitions between relative income categories and by examining dynamics of proximate determinants separately.
- Microeconomic priorities: study firm-level learning and technological upgrading in developing countries.
- Broader supply-side factors: variation in provision and quality of infrastructure may contribute to persistent low aggregate TFP.
- Consideration of technological and structural shifts:
  - The ongoing AI revolution “may disproportionately benefit advanced economies, while proving harder to apply in middle-income countries” (Alonso et al. 2020).
  - Impact of a green transition is ambiguous.
- Policy takeaway: greater understanding of TFP dynamics, firm upgrading, infrastructure quality, and implications of new technologies is likely necessary for policymakers concerned about middle-income traps.

### Selected methods appendix points
- Uniqueness of stationary distributions: use Dobrushin coefficient α(p) to check global stability; process is globally stable if there exists strictly positive integer t such that α(p^t) > 0.
- Mean passage times: computed using the fundamental matrix Z ≡ (I − P + W)^{-1} where W rows equal the stationary distribution; mean passage time from i to j (i ≠ j) given by m_ij = (z_jj − z_ij) / w_j.

*Source: Annex I. Quintiles and mobility, wpiea2024091-print-pdf*

### Annex I. Quintiles and mobility ........................................................................................

### Annex I. Quintiles and mobility

### Unique Stationary Distributions
- Methodology:
  - Use finite state Markov chains to study mobility, drawing on ideas in Quah (1993), Feyrer (2008), and Im and Rosenblatt (2015).
  - Employ transition probability matrices; when the transition probability from middle-income to high-income is low compared to other upward transitions, that suggests greater difficulty achieving high-income status (a “middle-income trap” interpretation).
  - Complement transition analysis with mean passage times derived from transition matrices to capture the influence of both upwards and downwards mobility.
  - Systematic use of “constant growth thresholds” to define income categories; argued to be the default choice for middle-income trap analysis.
- Empirical scope:
  - Long-span GDP data from 1950 to 2015 (65 years), using five-year intervals, yielding more than 1,800 transitions between income categories.
  - Sample represents more than 93 percent of the world population in 2018.
  - Additional analysis using Penn World Table (PWT) version 10, covering 1975–2015.
- Key high-level findings related to stationary distributions:
  - Mobility within the distribution of output per head is low.
  - There is no long-run tendency for absolute convergence in output per head.
  - Little evidence that transitions from a middle-income category are uniquely difficult (i.e., weak support for a distinct middle-income bottleneck in output dynamics alone).

### Mean Passage Times
- Rationale and metrics:
  - Mean passage times provide the average time to move between particular states, conditioning on the possibility of both upward and downward moves.
  - A continuous summary statistic is proposed, derived by comparing mean passage times under constant growth thresholds, to quantify the extent of a trap rather than using a binary existence test.
- Empirical implications drawn in the paper:
  - When applied to GDP per head transitions (1950–2015 and 1975–2015 PWT), mean passage times and transition matrices show low mobility and no strong evidence of uniquely long middle-income persistence in output per head.
  - However, analyses of proximate determinants of GDP per head reveal a different picture:
    - Relative capital intensity and human capital show upwards mobility and convergence across countries.
    - Relative TFP shows a lack of convergence; this divergence in TFP has been masked by convergence in capital intensity and human capital.
    - If current patterns persist, future outcomes for output will be driven mainly by the TFP process.
  - Consequently, a middle-income trap may become increasingly apparent in the years to come unless:
    - Higher human capital helps close gaps in relative TFP, or
    - The convergence process has changed in recent decades (as argued elsewhere).
- Policy-relevant interpretation:
  - The notion of a trap is operationalized as when emerging from middle income is a lengthy and uncertain process compared unfavorably with historical precedents.
  - Mean passage times and the continuous measure allow comparison of how protracted escape from middle income would be under prevailing transition probabilities.

### Countries Used
- Data and samples mentioned:
  - Main GDP analysis: countries covering more than 93 percent of the world population in 2018, using 1950–2015 Maddison data with five-year transitions (more than 1,800 transitions).
  - Complementary analysis: Penn World Table (PWT) version 10, 1975–2015 sample.
- Related dataset notes (as listed in the source):
  - Figures and tables index includes:
    - Figure 1: The Stationary Distributions.
    - Tables covering five-year transitions and mean passage times for Maddison data and PWT (multiple tables listed for GDP per capita, K/Y, human capital, TFP).
    - Country lists referenced in tables: e.g., “145 Countries Used in Table 1”, “134 Countries Used in Table 3”, “122 Countries Used in Tables 11–13”.
- Scope caution:
  - Analyses assume that transition probabilities remain constant over time for the purpose of extrapolating long-run implications from estimated transition matrices.

### Main contributions (as summarized in the source)
- Explain why constant growth thresholds should be the default choice for Markov chains applied to the middle-income trap question.
- Replace a yes/no dichotomy for the existence of a trap with a continuous summary measure of its extent (based on mean passage times under constant growth thresholds).
- Study potential traps not only in output dynamics but in the proximate determinants of output per head (capital intensity, human capital, TFP).
- Take advantage of longer time spans of data now available (e.g., 1950–2015) to obtain precise estimates of transition probabilities.

*Source: Annex I. Quintiles and mobility, wpiea2024091-print-pdf*

### 1989. Since our samples typically end in 2015, we bring more than twenty-five years of new data to bear on

### wpiea2024091-print-pdf - 1989. Since our samples typically end in 2015, we bring more than twenty-five years of new data to bear on

### Background
- Distinguishes between an absolute trap (income range above which growth becomes more difficult) and a relative trap (range of income relative to the frontier that is hard to move beyond). The paper addresses only the relative trap.
- The relative-trap empirical strategy builds on Quah (1993) using finite-state Markov chains and transition probabilities between relative income classes.
- The authors adopt constant growth thresholds so that a country growing at a constant relative rate takes a constant length of time to traverse each intermediate state; thresholds therefore increase geometrically.
- The paper introduces a summary statistic ∆MPT: the difference in mean passage times between the upwards transition to the highest category and the upwards transition to the next-highest category, used to quantify the extent of a middle-income trap.
- Key prior findings cited:
  - Factor accumulation (capital, human capital) drove East Asian growth, with relatively limited TFP gains (Krugman 1994, Young 1995).
  - Some literature found limited evidence for a distinct middle-income trap when examining transition probabilities alone (Quah 1993; Kremer et al. 2001).

### Methods
- Uses finite-state Markov chains on five-year intervals (following Kremer et al. 2001) to estimate transition matrices P = [pij] where pij = P{Xn = j | Xn−1 = i}.
- Marginal distribution evolves as ψt+1 = ψt P; stationary distribution ψ∗ satisfies ψ∗ = ψ∗ P.
- Reports transition counts as well as transition probabilities; asymptotic standard errors for estimated pij are reported using Anderson and Goodman (1957).
- Convergence speed is summarized by γ, the number of periods needed to halve the norm of the difference between the current distribution and the stationary distribution, based on the second largest eigenvalue λ2 of P. γ is adjusted for five-year intervals.
- Five-year intervals chosen because annual data often violate the Markov property; ten-year matrices compared with the square of five-year matrices as a check.

### Data
- Two primary data sources:
  - Maddison Project Database 2020: allows starting the analysis in 1950; balanced panel of 145 countries observed at 14 points (five-year intervals) → NT = 145×(14−1) = 1,885 transitions.
  - Penn World Table (PWT) version 10.01 (‘rgdpo’): typical sample starts in 1975 and ends in 2015; when using PWT, real GDP per adult of working age (15-64) is used (working-age population from World Development Indicators).
- Sample coverage:
  - Largest sample represents more than 7.1 billion people in 2018, or more than 93 percent of that year’s total population.
  - PWT sample used in results represent 92.7 percent of 2018 global population (134 countries, T = 8 five-year transitions per country, NT = 1,072 transitions).
- Primary constant growth thresholds used for income categories (relative to the U.S.): (0.09, 0.18, 0.36, 0.72).
  - Middle-income explicitly defined in many analyses as between 36% and 72% of the US GDP per head.
- For proximate determinants (K/Y, human capital h, and TFP A), thresholds used in that section are (0.32, 0.48, 0.72), which increase geometrically by factor 3/2, retaining 0.72 as the advanced-economy benchmark.

### Results — Aggregate GDP per Head (Maddison and PWT)
- Maddison (1950–2015; 145 countries; NT = 1,885):
  - Upwards mobility is slow and there are meaningful downwards transitions; stationary distribution places greatest mass on the highest output category (>72% of U.S.), yet countries will spend about two-thirds of their time below this category (so no absolute convergence).
  - Overall evidence is for limited mobility rather than a sharply distinct middle-income trap when looking only at transition probabilities.
- PWT (1975–2015; 134 countries; NT = 1,072; T = 8):
  - Transition matrix (Table 3) summary (five-year transitions, PWT):
    - Row/column order: <0.09, <0.18, <0.36, <0.72, <∞.
    - Selected transition probabilities (with asymptotic std. errors in parentheses as reported):
      - From <0.09 to <0.09: 0.909 (0.015)
      - From <0.09 to <0.18: 0.086 (0.015)
      - From <0.18 to <0.09: 0.167 (0.026)
      - From <0.18 to <0.18: 0.685 (0.033)
      - From <0.36 to <0.36: 0.770 (0.030)
      - From <0.72 to <0.36: 0.777 (0.031)
      - From <∞ to <0.36: 0.889 (0.027)
    - Transition counts (Table 3):
      - Row <0.09: 318, 30, 2, 0, 0
      - Row <0.18: 34, 139, 30, 0, 0
      - Row <0.36: 1, 23, 154, 22, 0
      - Row <0.72: 0, 2, 16, 143, 23
      - Row <∞: 1, 14, 120, 0, 0
    - Convergence statistic: γ = 82.6 (five-year periods)
    - Marginal distributions:
      - Last period: 0.276, 0.172, 0.209, 0.164, 0.179
      - +25 years: 0.289, 0.158, 0.190, 0.177, 0.186
      - +100 years: 0.290, 0.151, 0.179, 0.181, 0.199
      - Stationary ψ∗: 0.280, 0.148, 0.179, 0.185, 0.208
  - Mean passage times (Table 4, PWT; times in years, rounded to one decimal place):
    - From <0.09 to:
      - <0.18: 62.5
      - <0.36: 138.6
      - <0.72: 283.9
      - <∞: 445.4
    - From <0.18 to:
      - <0.09: 133.0
      - <0.36: 89.5
      - <0.72: 234.8
      - <∞: 396.3
    - From <0.36 to:
      - <0.09: 249.9
      - <0.18: 124.3
      - <0.72: 145.3
      - <∞: 306.8
    - From <0.72 to:
      - <0.09: 338.0
      - <0.18: 211.7
      - <0.36: 109.2
      - <∞: 161.5
    - From <∞ to:
      - <0.09: 377.1
      - <0.18: 250.9
      - <0.36: 147.0
      - <0.72: 54.7
    - Mean first recurrence times (MFR) by state: 17.8, 33.8, 28.0, 27.0, 24.0
  - Interpretation:
    - Stationary distribution for output per head is relatively flat; more than half the world’s countries are below the highest category in the stationary distribution.
    - Little evidence for a pronounced middle-income trap in transition probabilities alone; rather, evidence points to limited overall mobility and slow convergence.

### Results — Proximate Determinants (previewed in text)
- Approach: Use Cobb-Douglas framework Y = A K^β (hL)^{1−β} and express in output per working-age adult and capital-output ratio to study distributions and transitions of relative K/Y, human capital h, and TFP A.
- Thresholds for proximate determinants: (0.32, 0.48, 0.72) (geometric factor 3/2).
- Key high-level finding summarized in the text:
  - Capital intensity and human capital are converging across countries.
  - Relative TFP convergence is much slower; the lack of TFP convergence is likely to begin to dominate outcomes for relative output per head if current patterns continue.
  - The combination implies that convergence via factor accumulation may exhaust before countries reach rich-country status, increasing the relevance of a middle-income outcome in the future.

### Robustness and inference considerations
- Reporting of transition counts helps gauge which transition probabilities are well estimated; low counts for a well-observed source state do not necessarily imply imprecision for the associated pij.
- Slow convergence of GDP per head implies stationary distributions can be sensitive to small changes in transition probabilities; by contrast, proximate determinants typically converge faster and their stationary distributions are more robust.
- Benchmarks: Primary analysis measures outcomes relative to the U.S.; robustness includes measuring relative to the median of the G7.
- Choice of thresholds matters; the authors consider alternative threshold choices in robustness checks (e.g., World Bank thresholds at market exchange rates are not used because PPP comparisons are preferred here).

*Source: IMF working paper content provided in the supplied PDF excerpt.*

### 2015. There is substantial upwards mobility in this proximate growth determinant, but only limited

### wpiea2024091-print-pdf - 2015. There is substantial upwards mobility in this proximate growth determinant, but only limited

### Capital intensity (K/Y): transitions, convergence, and mean passage times
- Five-year transition matrix, K/Y, 1975–2015 (probabilities and standard errors in parentheses):
  - Row <0.32: 0.660 (0.037), 0.241 (0.034), 0.093 (0.023), 0.006 (0.006)
  - Row <0.48: 0.047 (0.015), 0.524 (0.036), 0.340 (0.034), 0.089 (0.021)
  - Row <0.72: 0.004 (0.004), 0.101 (0.018), 0.552 (0.030), 0.343 (0.029)
  - Row <∞ : 0.005 (0.003), 0.049 (0.009), 0.946 (0.009)
- Convergence parameter: γ = 12.6
- Distributional snapshots:
  - Last period: 0.000, 0.058, 0.129, 0.813
  - +25 years: 0.006, 0.039, 0.124, 0.831
  - +100 years: 0.007, 0.038, 0.122, 0.834
  - Stationary ψ*: 0.007, 0.038, 0.122, 0.834
- Transition counts and sample:
  - NT = 1240, N = 155, T = 8
- Mean Passage Times (MPT) and Mean First Return (MFR), K/Y (times in years, rounded to one decimal place):
  - MPT matrix:
    - From <0.32 to <0.32: 0.0; to <0.48: 95.3; to <0.72: 39.3; to <∞: 38.5
    - From <0.48 to <0.32: 2177.9; to <0.48: 0.0; to <0.72: 32.2; to <∞: 26.8
    - From <0.72 to <0.32: 2390.1; to <0.48: 272.6; to <0.72: 0.0; to <∞: 17.5
    - From <∞  to <0.32: 2463.2; to <0.48: 340.3; to <0.72: 95.4; to <∞: 0.0
  - MFR row: 765.8, 132.6, 41.1, 6.0
- Interpretation:
  - Substantial upwards mobility in capital intensity, limited downwards mobility.
  - Distribution by 2010–2015 already close to stationary distribution for capital intensity.
  - Unlikely that an equilibrium-based middle-income trap is explained by capital intensity differences.

### Human capital: transitions and mean passage times
- Human capital index constructed as exp(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 s > 8
- Five-year transition matrix, Human Capital, 1975–2015:
  - Row <0.32: 0.774 (0.075), 0.226 (0.075)
  - Row <0.48: 0.007 (0.005), 0.856 (0.020), 0.137 (0.020)
  - Row <0.72: 0.010 (0.005), 0.893 (0.015), 0.097 (0.015)
  - Row <∞ : 0.014 (0.007), 0.986 (0.007)
- Convergence parameter: γ = 36.2
- Distributional snapshots:
  - Last period: 0.000, 0.186, 0.357, 0.457
  - +25 years: 0.003, 0.100, 0.309, 0.588
  - +100 years: 0.001, 0.026, 0.179, 0.794
  - Stationary ψ*: 0.000, 0.009, 0.122, 0.869
- Transition counts and sample:
  - NT = 1032, N = 129, T = 8
- Mean Passage Times, Human Capital (times in years, rounded to one decimal place):
  - MPT matrix:
    - From <0.32: to <0.32: 0.0; to <0.48: 22.1; to <0.72: 59.6; to <∞: 114.9
    - From <0.48: to <0.32: 86536.9; to <0.48: 0.0; to <0.72: 37.5; to <∞: 92.7
    - From <0.72: to <0.32: 90621.2; to <0.48: 4084.4; to <0.72: 0.0; to <∞: 55.3
    - From <∞ : to <0.32: 90988.7; to <0.48: 4451.9; to <0.72: 367.5; to <∞: 0.0
  - MFR row: 19545.6, 565.7, 41.1, 5.8
- Interpretation:
  - Very little downwards mobility in relative human capital, some upwards mobility but slow into the highest category.
  - Stationary distribution concentrates most mass in highest category; absolute convergence in human capital is expected eventually.
  - Even after another 25 years, about 40 percent of countries remain outside the highest category of relative human capital.

### Total Factor Productivity (TFP): PWT measure, transitions, and mean passage times
- TFP measure used: ‘ctfp’ from version 10.01 of the Penn World Table (PWT), 1975–2015.
- Five-year transition matrix, TFP from PWT, 1975–2015:
  - Row <0.32: 0.750 (0.082), 0.250 (0.082)
  - Row <0.48: 0.087 (0.029), 0.685 (0.048), 0.217 (0.043), 0.011 (0.011)
  - Row <0.72: 0.011 (0.008), 0.173 (0.028), 0.631 (0.036), 0.184 (0.029)
  - Row <∞ : 0.005 (0.003), 0.126 (0.016), 0.869 (0.016)
- Convergence parameter: γ = 23.8
- Distributional snapshots:
  - Last period: 0.033, 0.200, 0.322, 0.444
  - +25 years: 0.073, 0.211, 0.278, 0.438
  - +100 years: 0.090, 0.227, 0.274, 0.409
  - Stationary ψ*: 0.092, 0.229, 0.274, 0.405
- Transition counts and sample:
  - NT = 720, N = 90, T = 8
- Mean Passage Times, PWT TFP (times in years, rounded to one decimal place):
  - MPT matrix:
    - From <0.32: to <0.32: 0.0; to <0.48: 20.0; to <0.72: 51.4; to <∞: 102.1
    - From <0.48: to <0.32: 197.4; to <0.48: 0.0; to <0.72: 31.4; to <∞: 82.1
    - From <0.72: to <0.32: 248.9; to <0.48: 64.3; to <0.72: 0.0; to <∞: 55.2
    - From <∞ : to <0.32: 285.3; to <0.48: 100.2; to <0.72: 39.4; to <∞: 0.0
  - MFR row: 54.3, 21.8, 18.3, 12.3
- Interpretation:
  - Upwards mobility exists but with noticeable downwards mobility, especially from the highest category.
  - Stationary distribution places significant mass on the second-highest category.
  - Analysis predicts that, even in the very long run, more than half the world’s countries will have TFP less than 72 percent of the U.S. level.
  - Lack of absolute convergence in TFP may be one source of a middle-income trap; sample limitation noted (90 countries, ~78 percent of 2018 world population).

### Synthesis: implications of proximate GDP determinants for the middle-income trap
- Key comparative result:
  - Lack of absolute convergence in relative TFP contrasts with apparent absolute convergence in capital-output ratio and human capital.
  - Relative capital-output ratio distribution is already close to stationary; human capital converges toward the highest state.
- Implication:
  - Because TFP fails to converge absolutely, a middle-income trap may become more prominent over time; many countries may remain well short of the frontier even as capital intensity and human capital converge to high levels.
  - The failure to achieve absolute convergence in relative TFP will leave many countries well short of the frontier.

### Common sample analysis (basic TFP, shared sample, and G7 benchmarking)
- Motivation and changes:
  - Use a common sample representing 92 percent of the 2018 global population.
  - Replace sophisticated PWT TFP with a ‘basic’ TFP computed from equation (1), requiring an assumption about output-capital elasticity β.
  - Feenstra et al. (2015) report an average labor share of 0.52 implying β = 0.48 under perfect competition (this value is higher than usually adopted; alternate β considered elsewhere in the paper).
  - Focus series on output per working-age adult and proximate determinants, with four-state Markov chains.
- Conceptual note:
  - Output per head is a convex function of TFP conditional on the capital-output ratio, so stationary distributions of relative output per head will not simply mirror relative TFP distributions.
- Statistic for middle-income trap: ∆MPT = absolute difference in mean passage times between the upwards transition to the highest category and the upwards transition to the next-highest category.
- Findings, Common Sample (benchmarked to US):
  - Table 11 stationary distributions and ∆MPT (NT = 976, N = 122, T = 8):
    - Output per w/a adult: 0.619, 0.076, 0.107, 0.198; ∆MPT = -24.8
    - Basic TFP: 0.224, 0.383, 0.255, 0.138; ∆MPT = 127.2
    - K/Y ratio: 0.006, 0.040, 0.134, 0.820; ∆MPT = -11.2
    - Human capital: 0.000, 0.006, 0.099, 0.894; ∆MPT = 17.2
  - Interpretation:
    - Strong lack of upwards mobility in (basic) TFP: ∆MPT = 127.2 years implies moving into highest relative TFP category may take many decades on average.
    - Upwards mobility in capital intensity and human capital offsets basic TFP stagnation for output per working-age adult, producing less pronounced evidence of a middle-income trap in output per working-age adult.
- Findings, Common Sample, Benchmarked to G7 median:
  - Table 12 stationary distributions and ∆MPT (NT = 976, N = 122, T = 8):
    - Output per w/a adult: 0.638, 0.104, 0.070, 0.188; ∆MPT = 81.4
    - Basic TFP: 0.078, 0.208, 0.314, 0.401; ∆MPT = 29.6
    - K/Y ratio: 0.029, 0.078, 0.216, 0.677; ∆MPT = 1.8
    - Human capital: 0.000, 0.004, 0.131, 0.866; ∆MPT = 13.3
  - Interpretation:
    - Benchmarking against the G7 median changes the magnitudes and some assessments: more evidence of a middle-income trap for output, and somewhat less evidence for basic TFP (∆MPT for basic TFP falls to 29.6).
    - Results highlight sensitivity of trap metrics to choice of benchmark (US vs. G7 median).

*International Monetary Fund. Content unit: wpiea2024091-print-pdf - 2015. There is substantial upwards mobility in this proximate growth determinant, but only limited*

### 29.6 years. But it remains the case that, even in the long run, almost 60 percent of countries will be

### At the Threshold: The Increasing Relevance of the Middle-Income Trap

### Alternative thresholds and key distributional findings (Table 13)
- Alternative thresholds used: <0.36, <0.48, <0.64, < ∞.
- Stationary distribution (common sample, new thresholds) for:
  - Output per w/a adult: 0.639, 0.058, 0.067, 0.236; ∆ MPT = -9.7
  - Basic TFP: 0.361, 0.287, 0.170, 0.182; ∆ MPT = 71.7
  - K/Y ratio: 0.014, 0.034, 0.088, 0.864; ∆ MPT = -20.6
  - Human capital: 0.000, 0.002, 0.031, 0.967; ∆ MPT = 3.2
- Sample dimensions reported: NT = 976; N = 122; T = 8.
- Sensitivity note: using a lower β = 1/3 yields a mid-level trap for relative TFP as well, with even less upwards mobility into the top category.

### Interpretation of these results
- The alternative (less demanding) highest threshold (0.64 vs 0.72) produces broadly similar stationary distributions, but the ∆MPT statistic for relative TFP falls to 71.7 years.
- The authors infer that capital intensity and human capital appear on course for absolute convergence, while relative TFP exhibits persistent lack of upwards mobility into the highest category.
- Projection: “Over time, a middle-income trap for output per head may become more apparent rather than less, once the convergence of capital intensity and human capital no longer offsets the lack of upwards mobility into advanced levels of relative TFP.”

### Robustness: first-order Markov assumption and time aggregation (Table 14)
- Method: compare 10-year transitions with the square of the 5-year transition matrix (Maddison data, 1950–2010).
- Main observation: the two matrices are “remarkably similar,” with the main difference being more downwards transitions from the highest output category in 5-year transitions (some may be temporary).
- Implied stationary distributions:
  - Stationary (5-year) ψ*: 0.129, 0.095, 0.167, 0.237, 0.371
  - Stationary (10-year) ψ*: 0.096, 0.076, 0.141, 0.234, 0.453
- Sample dimensions reported: N = 145; T = 6 (10-year); T = 12 (5-year).
- Conclusion: results support use of a first-order process for 5-year intervals on the Maddison data; similar conclusions hold for 10-year transitions in Penn World Table data (results not reported).

### Robustness: exclusion of oil/mineral exporters and subperiods (Table 15)
- Comparison of stationary distributions for the Maddison data:
  - With oil/mineral producers vs Without:
    - <0.09: 0.130 vs 0.129
    - <0.18: 0.100 vs 0.090
    - <0.36: 0.174 vs 0.142
    - <0.72: 0.248 vs 0.265
    - < ∞: 0.348 vs 0.374
- Subperiod stationary distributions (Maddison data):
  - 1950–1980: 0.140, 0.101, 0.205, 0.224, 0.331
  - 1980–2015: 0.182, 0.145, 0.191, 0.230, 0.252
- Interpretation:
  - Excluding the 20 Kremer et al. (2001) oil/mineral producers leaves stationary distributions little changed.
  - No clear tendency for 1980–2015 to indicate greater long-run absolute convergence than 1950–1980 using these methods; a subperiod starting in the 1990s or early 2000s might yield different evidence but would need more years of data for reliable findings.

### Conclusion: synthesis of empirical findings
- Stationary distribution for output per head implies long-run dispersion across relative income categories—consistent with absence of absolute convergence.
- Mean passage times generally argue against a distinct middle-income trap; instead, there is a general lack of mobility and failure to achieve absolute convergence.
- Crucial extension: proximate determinants show divergent dynamics:
  - Capital-output ratios and human capital: on course for absolute convergence (hiding lack of convergence in aggregate).
  - Relative TFP: predicted to lack upwards mobility and to dominate long-run dynamics—this could make middle-income traps increasingly apparent if patterns persist.
- Qualifications:
  - Results extrapolate from experience since the mid-1970s; convergence processes may have changed in the most recent twenty years.
  - Higher absolute levels of human capital might enable future upwards mobility in relative TFP (to be investigated in Imam and Temple (2024)).

### Policy implications and research priorities
- Empirical approach recommendation: study the middle-income trap using transitions between relative income categories and by examining dynamics of income determinants separately.
- Microeconomic focus: importance of studying firm-level learning and technological upgrading in developing countries (review reference: Verhoogen (2023)).
- Broader supply-side factors: variation in provision and quality of infrastructure may contribute to persistent low aggregate TFP.
- Consideration of technological and structural shifts:
  - The ongoing AI revolution “may disproportionately benefit advanced economies, while proving harder to apply in middle-income countries” (Alonso et al. 2020).
  - The impact of a green transition is ambiguous in its effects.
- Policy takeaway: greater understanding of TFP dynamics, firm upgrading, infrastructure quality, and the implications of new technologies is likely necessary for policymakers concerned about middle-income traps.

### Methods appendix (selected technical points)
- Uniqueness of stationary distributions: use Dobrushin coefficient α(p) to check global stability; process is globally stable if there exists strictly positive integer t such that α(p^t) > 0.
- Mean passage times: computed using the fundamental matrix Z ≡ (I − P + W)^{-1} where W rows equal the stationary distribution; mean passage time from i to j (i ≠ j) given by m_ij = (z_jj − z_ij) / w_j.

*International Monetary Fund — At the Threshold: The Increasing Relevance of the Middle-Income Trap.*

### 84. American Journal of Political Science, 50(3), 551-569.

### 84. American Journal of Political Science, 50(3), 551-569.

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*wpiea2024091-print-pdf - 84. American Journal of Political Science, 50(3), 551-569. Canonical URL: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024091-print-pdf.pdf*

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