A Non-unitary Discount Rate Model

We develop a simple macroeconomic model in which agents discount their utility from consumption and the utility from leisure at different rates. Under this setting, time-inconsistency emerges for the preferences of agents. Moreover, the time-inconsistency problem generates two types of inefficiencies: intratemporal and intertemporal. We examine the welfare effects of savings subsidy and consumption tax. The effects of taxation in our model are quite different from those in the standard model. If the discount rate for consumption is higher (lower) than that for leisure, today's self cares less (more) about the consumption of the future selves than the leisure of the future selves. Depressing (stimulating) the consumption of future selves improves the utility of today's self. Hence a positive (negative) consumption tax rate improves the utility level of all selves.