Over time, both productivity and the GDP per capita have increased in industrialized countries. Interestingly, the productivity and the GDP per capita of these nations have approached one another over time. This convergence signifies that all industrialized nations are approaching a common level of prosperity.
Why does this phenomenon occur? Nobody has a complete answer to this question. The most common explanation for convergence is the constantly increasing speed at which new technologies spread across international borders. Remember that one of the keys to increased productivity is technological improvement. When industrialized nations share technological advances (rather than forcing each country to make them independently), productivity for each country will tend to move towards a level dictated by the performance of modern production technology.
We do not see convergence in all countries. Convergence only occurs among the industrialized nations. These are countries that have the infrastructure, government, and education level to utilize the technological advances that can potentially their production capabilities. Countries that lack a solid infrastructure, possess an unstable government, or do not possess an educated populace are unable to benefit from the technological advances that are enjoyed by the industrialized countries and are thus not covered under the idea of convergence.
This can be illustrated clearly. Say two countries produce widgets. One country makes widgets by hand at a rate of 1 per day. The other uses advanced machines to produce widgets at a rate of 100 per day. A technological advance allows the widget making machine to begin producing 1000 widgets per day. Unfortunately, only the country that has the widget machine is able to benefit from this technological advance. The other is still only able to sustain the 1 widget per day level. In this example, the industrialized nations that do converge are like the mechanized widget producer while the non-industrialized countries that do not converge are like the non-mechanized widget producer.