We study testable implications for the dynamics of consumption and income of models in which first best allocations are not achieved because of a moral hazard problem with hidden saving. We show that in this environment agents typically achieve more insurance than that obtained under self insurance with a single asset. Consumption allocations exhibit 'excess smoothness' as found and defined by Campbell and Deaton (1989). We argue that excess smoothness, in this context, is equivalent to a violation of the intertemporal budget constraint considered in a Bewley economy (with a single asset). We also show parametrizations of our model in which we can obtain a closed form solution for the efficient insurance contract and where the excess smoothness parameter has a structural interpretation in terms of the severity of the moral hazard problem. We present tests of excess smoothness, applied to UK micro data, and constructed using techniques proposed by Hansen et al. (1991) to test the intertemporal budget constraint. Our theoretical model leads to interpret them as tests of the market structure faced by economic agents. We also construct a test based on the dynamics of the cross sectional variances of consumption and income that is, in a precise sense, complementary to that based on Hansen et al (1991) and that allows to estimate the same structural parameter. The results we report are consistent with the implications of the model and internally coherent.