The tendency of sectoral demand to satiate has long been argued to be a key driver of the structural change in an economy (Pasinetti 1981; Saviotti 2001). This literature raises the question as to what extent cross-sectional patterns of household expenditure can be used to make inferences about how the demand for goods and services will grow over time. Moreover, if indeed satiation does take place, then firms and entrepreneurs could react to this situation by innovating goods and services in order to overcome stagnation in demand growth (Witt 2001). We empirically investigate this 'satiation-escape' hypothesis by examining the inter-temporal dynamics of Engel curves and their derivatives, which reflect how household spending on a good changes with income. Taking into account changes in the price level, we investigate whether Engel curves that exhibit cross-section satiation tend to exhibit over time an upwards shift in the satiation level jointly with a shift in position and shape. Evidence suggests that this is actually the case.