It has been suggested that the relationship between income and the associated environmental pressure in terms of derived materials inputs and pollution levels may take an inverted-U shape, indicating a 'delinking' of environmental pressure from economic growth in relation to rising per capita incomes. The likelihood of such a relationship being persistent is discussed in the context of a simple macro model of industrial metabolism, and the possibility of 'relinking' clearly emerges. Data on specific indicators of environmental pressure (i.e., the throughput of materials, energy and the volume of transport) in 19 countries have been used to investigate patterns of delinking. The results suggest that delinking may not be persistent; already some advanced economies may be entering a new period of relinking. The relationships between environmental pressure and welfare in the medium long term may be N-shaped rather than inverted-U-shaped.