This paper characterizes the transition dynamics of a continuous-time neoclassical production economy with capital accumulation in which households face idiosyncratic income risk and cannot commit to repay their debt. Therefore, even though a full set of contingent claims that pay out conditional on the realization of idiosyncratic shocks is available, the equilibrium features imperfect insurance and a non-degenerate crosssectional consumption distribution. When household labor productivity takes two values, one of which is zero, and the utility function is logarithmic, we characterize the entire transition dynamics induced by unexpected technology shocks, including the evolution of the consumption distribution, in closed form. Thus, the model constitutes an analytically tractable alternative to the standard incomplete markets general equilibrium Aiyagari (1994) model by retaining its physical environment, but replacing the incomplete asset markets structure with one in which limits to consumption insurance emerge endogenously due to limited commitment.