Testing Absolute Convergence: The Solow Model’s Key Prediction

Empirical Evidence from the Penn World Tables

Author
Affiliation

Claudius Gräbner-Radkowitsch

EUF and JKU

Introduction

The Solow-Swan model makes one of the most striking predictions in development economics: absolute convergence. According to this model, poor countries should grow faster than rich countries because of diminishing returns to capital, leading all economies to eventually converge to the same level of income per capita.

This prediction emerges from two key assumptions:

  1. Diminishing returns to capital: As countries accumulate more capital, additional units of capital become less productive
  2. Factor mobility: Capital and technology can flow freely between countries

The mathematical intuition is straightforward: if the marginal product of capital is higher in poor countries (due to lower capital stock), then investment should flow there, generating higher growth rates until all countries reach the same steady state.

But does this prediction hold in reality? In this session’s exercise, you were asked to test the absolute convergence hypothesis using real-world data from the Penn World Tables. Below you find a possible solution for this exercise.

The Convergence Test

We test absolute convergence by estimating the following regression:

\[\frac{1}{T} \ln\left(\frac{y_{i,t+T}}{y_{i,t}}\right) = \alpha + \beta \ln(y_{i,t}) + \epsilon_i\]

Where:

  • \(\frac{1}{T} \ln\left(\frac{y_{i,t+T}}{y_{i,t}}\right)\) is the average annual growth rate of GDP per capita over period T
  • \(y_{i,t}\) is initial GDP per capita
  • \(\beta\) is the convergence parameter

Absolute convergence occurs if \(\beta < 0\): countries with lower initial income grow faster.

Data and Setup

We use data from the Penn World Tables 11.0, which provides internationally comparable GDP data using purchasing power parity (PPP) adjustments.

Packages used 
library(haven)
library(dplyr)
library(ggplot2)
library(tidyr)
library(broom)
library(knitr)
library(countrycode)

We first import the PWT data, select the years of interest and define the list of OECD countries.

Code 
# Load Penn World Tables data
pwt_data <- read_dta("pwt110.dta") %>%
  filter(year %in% c(1960, 1990, 2022), pop>1) %>%
  select(c("countrycode", "country", "year", "cgdpo"))

# Create OECD country list for subset analysis
oecd_countries <- c(
  "AUS", "AUT", "BEL", "CAN", "CHL", "CZE", "DNK", "EST", 
  "FIN", "FRA", "DEU", "GRC", "HUN", "ISL", "IRL", "ISR", 
  "ITA", "JPN", "KOR", "LVA", "LTU", "LUX", "MEX", "NLD", 
  "NZL", "NOR", "POL", "PRT", "SVK", "SVN", "ESP", "SWE", 
  "CHE", "TUR", "GBR", "USA")

# Display sample of data
kable(head(pwt_data, 5), caption = "Sample of Penn World Tables Data")
Sample of Penn World Tables Data
countrycode country year cgdpo
AGO Angola 1990 41716.39
AGO Angola 2022 225903.91
ALB Albania 1990 12227.74
ALB Albania 2022 43686.69
ARE United Arab Emirates 1990 300576.81

Then we create the data set used for the estimation by transforming it into the right shape, filter for missing observations and calculating the necessary variables (such as annual growth rates, initial GDP)

Code 
# Create convergence dataset
convergence_data <- pwt_data %>%
  pivot_wider(
    names_from = year,
    values_from = cgdpo, 
    names_prefix = "gdp_"
  ) %>%
  filter(
    !is.na(gdp_1960),
    !is.na(gdp_1990),
    !is.na(gdp_2022)
  ) %>%
  mutate(
    # Calculate annual growth rates (in decimal form)
    growth_60_90 = (log(gdp_1990) - log(gdp_1960)) / 30,
    growth_90_22 = (log(gdp_2022) - log(gdp_1990)) / 32,
    growth_60_22 = (log(gdp_2022) - log(gdp_1960)) / 62,
    
    # Initial GDP (log)
    ln_gdp_1960 = log(gdp_1960),
    ln_gdp_1990 = log(gdp_1990),
    
    # Create OECD indicator
    oecd = ifelse(countrycode %in% oecd_countries, "OECD", "Non-OECD")
  )

# Display summary statistics
kable(convergence_data %>% 
  select(
    country, gdp_1960, gdp_1990, 
    gdp_2022, growth_60_90, growth_90_22) %>%
  head(10), 
  digits = 3,
  caption = "Sample Countries: GDP Levels and Growth Rates")
Sample Countries: GDP Levels and Growth Rates
country gdp_1960 gdp_1990 gdp_2022 growth_60_90 growth_90_22
Argentina 71913.469 219936.453 1260669.75 0.037 0.055
Australia 178392.688 554314.625 1703845.00 0.038 0.035
Austria 78335.203 223089.672 581060.31 0.035 0.030
Burundi 2097.166 5712.411 13065.02 0.033 0.026
Belgium 104181.156 281506.406 665497.25 0.033 0.027
Benin 4094.783 7253.897 46482.14 0.019 0.058
Burkina Faso 5438.738 10643.562 52374.13 0.022 0.050
Bangladesh 94496.164 181639.328 1327167.88 0.022 0.062
Bolivia (Plurinational State of) 6382.087 17126.891 114655.68 0.033 0.059
Brazil 177550.828 1006764.188 3826745.25 0.058 0.042

Testing Absolute Convergence

We then test convergence for the different time horizons and country groups.

Table 1 summarizes the result of estimating the convergence regression for 1960-1990 for all countries (see also Figure 1).

Convergence regression for 1960-1990 (all countries) 
model_60_90_all <- lm(growth_60_90 ~ ln_gdp_1960, data = convergence_data)

# Extract results for the summary table below
results_60_90_all <- tidy(model_60_90_all)
r_squared_60_90_all <- glance(model_60_90_all)$r.squared

# Display results
kable(results_60_90_all, digits = 4, 
      caption = "Convergence Regression Results: All Countries (1960-1990)")
Table 1: Convergence regression for 1960-1990 (all countries)
Convergence Regression Results: All Countries (1960-1990)
term estimate std.error statistic p.value
(Intercept) 0.0361 0.0129 2.7999 0.0063
ln_gdp_1960 0.0003 0.0012 0.2773 0.7822
Code for Figure 1 
all_6090_plot <- ggplot(convergence_data, aes(x = exp(ln_gdp_1960), y = growth_60_90 * 100)) +
  geom_point(aes(color = oecd), alpha = 0.7, size = 3) +
  geom_smooth(method = "lm", se = TRUE, color = "#E63946", linewidth = 1.2) +
  scale_x_log10(labels = scales::dollar_format()) +
  scale_color_manual(values = c("OECD" = "#1f77b4", "Non-OECD" = "#ff7f0e")) +
  labs(
    title = "Test of Absolute Convergence: All Countries (1960-1990)",
    subtitle = paste0("β = ", round(results_60_90_all$estimate[2], 4), 
                     ", R² = ", round(r_squared_60_90_all, 3)),
    x = "GDP per capita in 1960 (log scale, PPP)",
    y = "Average Annual Growth Rate 1960-1990 (%)",
    color = "Country Group",
    caption = "Negative slope indicates convergence"
  ) +
  theme_minimal() +
  theme(
    plot.title = element_text(size = 14, face = "bold"),
    plot.subtitle = element_text(size = 12)
  )
all_6090_plot
Figure 1: Absolute Convergence Test: All Countries (1960-1990)

If we restrict ourselves to the subset of OECD countries we get the result summarized in Table 2 (see also Figure 2):

Convergence regression for 1960-1990 (OECD countries) 
# Filter for OECD countries
convergence_oecd <- convergence_data %>% filter(oecd == "OECD")

# Convergence regression for OECD countries
model_60_90_oecd <- lm(growth_60_90 ~ ln_gdp_1960, data = convergence_oecd)

# Extract results
results_60_90_oecd <- tidy(model_60_90_oecd)
r_squared_60_90_oecd <- glance(model_60_90_oecd)$r.squared

# Display results
kable(results_60_90_oecd, digits = 4, 
      caption = "Convergence Regression Results: OECD Countries (1960-1990)")
Table 2: Convergence regression for 1960-1990 (OECD countries)
Convergence Regression Results: OECD Countries (1960-1990)
term estimate std.error statistic p.value
(Intercept) 0.0669 0.0272 2.4626 0.0214
ln_gdp_1960 -0.0021 0.0023 -0.9337 0.3597
Code for Figure 2 
oecd_6090_plot <- ggplot(
  data = convergence_oecd, 
  mapping = aes(
    x = exp(ln_gdp_1960), 
    y = growth_60_90 * 100)
  ) +
  geom_point(color = "#1f77b4", alpha = 0.7, size = 3) +
  geom_smooth(method = "lm", se = TRUE, color = "#E63946", linewidth = 1.2) +
  scale_x_log10(labels = scales::dollar_format()) +
  labs(
    title = "Test of Absolute Convergence: OECD Countries (1960-1990)",
    subtitle = paste0("β = ", round(results_60_90_oecd$estimate[2], 4), 
                     ", R² = ", round(r_squared_60_90_oecd, 3)),
    x = "GDP per capita in 1960 (log scale, PPP)",
    y = "Average Annual Growth Rate 1960-1990 (%)",
    caption = "Negative slope indicates convergence"
  ) +
  theme_minimal() +
  theme(
    plot.title = element_text(size = 14, face = "bold"),
    plot.subtitle = element_text(size = 12)
  )
oecd_6090_plot
Figure 2: Absolute Convergence Test: OECD Countries (1960-1990)

Below you find the code for the other cases:

Convergence regression for 1990-2022 (all countries) 
# Convergence regression for 1990-2022 (all countries)
model_90_22_all <- lm(growth_90_22 ~ ln_gdp_1990, data = convergence_data)

# Extract results
results_90_22_all <- tidy(model_90_22_all)
r_squared_90_22_all <- glance(model_90_22_all)$r.squared

# Display results
# kable(results_90_22_all, digits = 4, 
#       caption = "Convergence Regression Results: All Countries (1990-2022)")
Code for Figure 3 
all_9022 <- ggplot(
  data = convergence_data, 
  mapping = aes(x = exp(ln_gdp_1990), y = growth_90_22 * 100)
  ) +
  geom_point(aes(color = oecd), alpha = 0.7, size = 3) +
  geom_smooth(method = "lm", se = TRUE, color = "#E63946", linewidth = 1.2) +
  scale_x_log10(labels = scales::dollar_format()) +
  scale_color_manual(values = c("OECD" = "#1f77b4", "Non-OECD" = "#ff7f0e")) +
  labs(
    title = "Test of Absolute Convergence: All Countries (1990-2022)",
    subtitle = paste0("β = ", round(results_90_22_all$estimate[2], 4), 
                     ", R² = ", round(r_squared_90_22_all, 3)),
    x = "GDP per capita in 1990 (log scale, PPP)",
    y = "Average Annual Growth Rate 1990-2022 (%)",
    color = "Country Group",
    caption = "Negative slope indicates convergence"
  ) +
  theme_minimal() +
  theme(
    plot.title = element_text(size = 14, face = "bold"),
    plot.subtitle = element_text(size = 12)
  )
Convergence regression for 1990-2022 (OECD countries) 
# Convergence regression for OECD countries
model_90_22_oecd <- lm(growth_90_22 ~ ln_gdp_1990, data = convergence_oecd)

# Extract results
results_90_22_oecd <- tidy(model_90_22_oecd)
r_squared_90_22_oecd <- glance(model_90_22_oecd)$r.squared

# Display results
# kable(results_90_22_oecd, digits = 4, 
#       caption = "Convergence Regression Results: OECD Countries (1990-2022)")
Code for Figure 4 
oecd9022_plot <- ggplot(
  data = convergence_oecd, 
  mapping = aes(x = exp(ln_gdp_1990), y = growth_90_22 * 100)
  ) +
  geom_point(color = "#1f77b4", alpha = 0.7, size = 3) +
  geom_smooth(method = "lm", se = TRUE, color = "#E63946", linewidth = 1.2) +
  scale_x_log10(labels = scales::dollar_format()) +
  labs(
    title = "Test of Absolute Convergence: OECD Countries (1990-2022)",
    subtitle = paste0("β = ", round(results_90_22_oecd$estimate[2], 4), 
                     ", R² = ", round(r_squared_90_22_oecd, 3)),
    x = "GDP per capita in 1990 (log scale, PPP)",
    y = "Average Annual Growth Rate 1990-2022 (%)",
    caption = "Negative slope indicates convergence"
  ) +
  theme_minimal() +
  theme(
    plot.title = element_text(size = 14, face = "bold"),
    plot.subtitle = element_text(size = 12)
  )

Summary and Interpretation

Summary of the findings

Table 3 provides a summary of our convergence test results.

Code for the summary table 
summary_results <- data.frame(
  Period = c("1960-1990", "1960-1990", "1990-2022", "1990-2022"),
  Sample = c("All Countries", "OECD Only", "All Countries", "OECD Only"),
  Beta = c(
    round(results_60_90_all$estimate[2], 4),
    round(results_60_90_oecd$estimate[2], 4),
    round(results_90_22_all$estimate[2], 4),
    round(results_90_22_oecd$estimate[2], 4)
  ),
  P_value = c(
    round(results_60_90_all$p.value[2], 4),
    round(results_60_90_oecd$p.value[2], 4),
    round(results_90_22_all$p.value[2], 4),
    round(results_90_22_oecd$p.value[2], 4)
  ),
  R_squared = c(
    round(r_squared_60_90_all, 3),
    round(r_squared_60_90_oecd, 3),
    round(r_squared_90_22_all, 3),
    round(r_squared_90_22_oecd, 3)
  ),
  Convergence = c(
    ifelse(
      results_60_90_all$estimate[2] < 0 & results_60_90_all$p.value[2] < 0.05, 
      "Yes", "No"),
    ifelse(
      results_60_90_oecd$estimate[2] < 0 & results_60_90_oecd$p.value[2] < 0.05, 
      "Yes", "No"),
    ifelse(
      results_90_22_all$estimate[2] < 0 & results_90_22_all$p.value[2] < 0.05, 
      "Yes", "No"),
    ifelse(
      results_90_22_oecd$estimate[2] < 0 & results_90_22_oecd$p.value[2] < 0.05, 
      "Yes", "No")
  )
)

kable(summary_results, 
      caption = "Summary of Convergence Test Results",
      col.names = c("Period", "Sample", "β (Convergence Parameter)", 
                   "P-value", "R²", "Significant Convergence?"))
Table 3: Summary of Convergence Test Results
Period Sample β (Convergence Parameter) P-value R² Significant Convergence?
1960-1990 All Countries 0.0003 0.7822 0.001 No
1960-1990 OECD Only -0.0021 0.3597 0.035 No
1990-2022 All Countries -0.0027 0.0581 0.040 No
1990-2022 OECD Only -0.0050 0.0078 0.260 Yes

Solow-Swan Prediction:

  • All countries should converge to the same income level due to diminishing returns to capital. Poor countries should grow faster than rich countries.

Our Results:

  • The evidence for absolute convergence is mixed at best and depends heavily on the sample
  • For a global sample, absolute convergence is not happening
  • The group of OECD countries show stronger convergence patterns than the global sample, but only for the more recent time period the convergence parameter is significantly smaller than zero
  • Even in this, the relationship is often weak (low R-squared values)

Why is it that absolute convergence is not happening as predicted by Solow-Swan? The great thing about such models is that we can systematically think about which of its assumptions might not hold:

  1. Different Technologies: Countries may not have access to the same production technologies
  2. Institutional Differences: Legal systems, property rights, and governance quality vary dramatically
  3. Human Capital: Education and skill levels differ substantially across countries
  4. Savings Rates: Cultural and policy differences affect investment rates
  5. Trade and Openness: Some countries are more integrated into global markets
  6. Geography and Natural Resources: Location and resource endowments matter
  7. Patterns of exploitation: Maybe the success of some countries comes at the expense of other countires that would then not converge for obvious reasons.

The weak evidence for absolute convergence has led economists to focus on conditional convergence - the idea that countries converge to their own steady states, which may differ based on:

  • Savings rates
  • Population growth rates
  • Institutional quality
  • Human capital levels
  • Technology access

As you see, this addresses some of the possible explanations above. Other economists, especially those working in Marxist and Dependency Theory traditions would be even more skeptical, and argue for a systematic relationship between development and underdevelopment, suggesting that studying conditional convergence is not a fruitful endeavor either.

Policy Implications

These findings have important implications for development policy:

  1. Investment Alone is Insufficient: Simply increasing capital investment won’t automatically generate catch-up growth
  2. Institutions Matter: Countries need good governance, property rights, and effective institutions
  3. Human Capital is Crucial: Education and skill development are essential for technology adoption
  4. Technology Transfer: Facilitating technology transfer and innovation is key for developing countries

Conclusion

Your empirical test of the Solow model’s convergence prediction illustrates the complexity of real-world development. While you found some evidence of convergence among similar countries (like OECD members), the prediction of universal convergence does not hold for the global economy.

This doesn’t mean the Solow model is wrong - rather, it highlights that the assumptions of identical technology, institutions, and preferences across countries are often violated in reality. The model remains valuable for understanding growth mechanics, but must be complemented with theories that account for institutional, technological, and structural differences between countries.

Understanding these limitations is crucial for development economists and policymakers who seek to design effective strategies for promoting economic growth and reducing global inequality.