This paper revisits the relationship between energy intensity and economic growth, using a flexible piecewise linear regression model. Based on a panel data set of 137 economies during 1990-2014, the analysis identifies a threshold effect of income growth on energy intensity change: although energy intensity is negatively correlated with income growth throughout the entire sample and study period, the declining rate significantly slows by more than 30 percent after the level of per capita income reaches ,000. Based on index decomposition, the analysis also finds that although structural change is important for intensity levels in all countries, the efficiency effect is more important in higher-income countries. The results suggest that when countries move beyond lower-middle-income levels, energy efficiency policies become far more critical for sustaining the rate of improvement in energy efficiency.