Lifecycle consumption and welfare with nonexponential discounting in continuous time

In a continuous-time lifecycle model with log utility and a general time-inconsistent discount function, we establish necessary and sufficient conditions under which commitment to the initial plan will increase the realized objective function for all future selves. We also establish necessary and sufficient conditions under which the log consumption profile over the lifecycle is locally concave. Empirically, the lifecycle profile of average household consumption is hump-shaped and thus concave at the peak, so this result is useful for identifying what discount functions are consistent with data. We express these conditions in terms of what we call the future weighting factor, which measures the deviation of the discount function from an (arbitrarily chosen) exponential discount function. Both the welfare and concavity conditions depend on how the marginal future weighting factor compares to a weighted average of marginal future weighting factors. If the marginal future weighting factor is sufficiently high at a given delay, i.e. if the discount function decays sufficiently more slowly than the chosen exponential at that delay, this implies that log consumption is concave at a point on the lifecycle profile and there will be a positive contribution to commitment utility relative to the realized utility for one of the selves.