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 cross-sectional 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. We then use these analytical transition results to study the speed of convergence in income per capita of a poor (low TFP) to a rich (high TFP) economy and the evolution of consumption inequality over time in response to an increase in idiosyncratic income risk.