Convergence or Catch-Up?

Are gaps in per capita income across countries narrowing? Are poor countries catching up with rich countries? These questions often come to mind when looking at patterns of economic growth across countries. At first glance, it is hard to tell the two questions apart. Both appear to refer to the same outcome of progress toward more equal living standards across countries, often referred to in common usage as “convergence”.

Upon further examination, these two questions turn out to have quite different answers (see Figure 1 plotting two measures from 1960 to 2019).

• To measure gaps in per capita income across countries, Figure 1 plots the mean log deviation (MLD) of per capita GDP across countries. The MLD is a common measure of inequality, which is used to measure the dispersion of per capita incomes across countries. Growth economists refer to declines in the dispersion of incomes across countries as “ σ -convergence”. For the remainder of this blog, I will refer to this as “convergence”, and increases in dispersion as “divergence”.¹ 
• To measure catching up, Figure 1 also shows the average across countries of GDP per capita expressed as a fraction of per capita GDP in the United States. Increases in this measure reflect a catching up of average living standards across all countries in the world to those of rich countries, as embodied by per capita GDP in the United States. I will refer to increases in this ratio as “catch-up”.

Figure 1 shows that “convergence” and “catch-up” are not the same thing — hardly the first time that the jargon of economists does not resonate well with common usage. Comparing 1960 and 2019, there was clear divergence because the dispersion of per capita incomes across countries is markedly higher in 2019 (MLD=0.67) than it was in 1960 (MLD=0.44). At the same time, there has clearly been catch-up because the average per capita incomes increased from 25 percent of US levels in 1960 to 35 percent in 2019.

Looking more closely at 20-year sub-periods, different combinations of convergence and catch-up can be seen. Between 1960–1979, there was divergence in increasing dispersion in per capita incomes, while at the same time average incomes relative to the US increased (divergence with catch-up). During the next 20-year period, dispersion in per capita incomes continued to increase, while average incomes relative to the USA were stagnant (divergence with no catch-up).  Finally, during 2000–2019, dispersion in per capita incomes fell slightly but average incomes relative to the USA increased sharply (convergence with catch-up).  These changing patterns of convergence are not new, see for example Kremer, Willis and Yang (2021) “Converging to Convergence” NBER Macroeconomics Annual, Figure 2 and Figure A5.

What accounts for the diverging patterns of convergence and catch-up over the past 60 years? The answer to this question lies in two facts about patterns of growth over this period.  

The first fact relates to the shifting patterns in the relationship between growth and initial income, which growth economists refer to as “ β -convergence” and applied microeconomists call “growth incidence curves” (see Figure 2). Each panel shows average annual growth over 20 years plotted against the log of initial per capita income, for the periods 1960-1979, 1980–1999, and 2000–2019. In the first two periods, the relationship is positive, — on average richer countries grew faster than poorer countries. In the last 20 years, the pattern has been reversed, — on average poorer countries have grown faster than richer countries. These changing patterns of β -convergence also are not new, see for example Kremer, Willis and Yang (2021) “Converging to Convergence” NBER Macroeconomics Annual, Figure 2 and Figure A5 as well as Patel, Sandefur and Subramanian (2021) “The New Era of Unconditional Convergence” Journal of Development Economics, Figure 1. 

The second fact relates to shifting patterns over time in average growth rates across countries relative to growth in the United States (see Table 1). During 1960–1979, the United States grew at almost exactly the same rate as the average growth rate of the world as a whole (2.7 percent versus 2.8 percent).  During 1980-1999, the United States grew considerably faster than the world as a whole (2.4 percent vs 1.4 percent), while in the last twenty years 2000-2019, the United States grew considerably slower (1.2 percent versus 2.4 percent).

The combination of these two facts accounts for the diverging patterns of convergence and catch-up in Figure 1. Consider first convergence, as measured by the trend in the mean log deviation of per capita income. Whenever growth is faster in richer countries than in poorer countries, the dispersion in per capita incomes increases (see appendix). This was the case during 1960–1979 and during 1980–1999 (see Figure 2). However, when growth is tilted towards poor countries, the mean log deviation decreases, as was the case during 2000-2019.²

Understanding patterns of catch-up hinges on an obvious point and a less obvious point. First, the obvious point: if average growth rates across all countries are higher than growth in the US, the average gap in incomes relative to the United States should fall. This is the case during 2000–2019, when average growth in the world was 2.4 percent per year as compared with just 1.3 percent in the United States, and there was clear catch-up. 

But what about the two other periods? Why did average per capita incomes relative to the United States only stagnate and not fall during 1980–1999 when growth in the United States was considerably faster than in the rest of the world? And why did average incomes relative to the United States increase during 1960–1979 even though the United States and the rest of the world grew at almost exactly the same rate during this time?  

Answering these questions depends on the less obvious point. Growth in average incomes relative to the United States is also sensitive to the “tilt” in growth rates across countries.  This is because a given growth rate applied to a larger base (in a richer country) results in a larger absolute increase in income, which in turn leads to a larger increase in overall average income than if the same growth had been applied to a smaller base (in a poor country). 

In summary, catch-up happens when growth is high relative to the United States, and/or when growth is tilted towards richer countries.  During the period 1960–1979, the United States and the rest of the world grew at the same rate on average. However, since growth was tilted towards rich countries during this period, average incomes relative to the United States increased, so there still was catch-up. Between 1980 and 1999, on average growth in the rest of the world lagged behind growth in the United States. However, this was offset by the “tilt” in growth rates towards richer countries, with the result that per capita incomes relative to the United States remained stagnant rather than declined. Finally, during 2000–2019, the modest tilt towards faster growth in poor countries was not enough to offset the relatively slow growth in the United States, leading to an increase in average incomes relative to the United States, so there was catch-up.

What does all this mean? The main lesson is that when discussing patterns of “convergence” — in the common usage sense of the word — in per capita incomes across countries, it is important to be precise about concepts under consideration. The terminology of convergence — understood to mean declining dispersion in per capita incomes across countries — and catch-up — understood to mean increases in average incomes relative to the United States — can be helpful in distinguishing between these two concepts, particularly when ostensibly similar measures move in opposite directions. It also is important to specify the time period under consideration, recognizing that different patterns of convergence and catch-up have been observed over the past sixty years. Table 2 offers a handy summary.

The Stata .do file that generates the results in this blog is available here. The file points to the Penn World Tables 10.0 version which is available here. This repository also includes code to reproduce the analysis using the Maddison Project Database with 145 countries since 1950 as well as an appendix deriving the results noted above.