Abstract: When discounting returns and costs at different points about the principal-agent problem are involved, nearly all existing studies choose exponential discounting function, which is the only discounting function standing for time-consistency. This paper relaxes the time-consistency preference hypothesis, taking quasi-hyperbolic discounting-βδ as the agents' discounting function. How principals inspire agents to work hard is considered in order to maximize their profits when principals' costs and returns occur at different points and returns are determined by the level of efforts and external factors. The findings suggest that the optimal contract is related with the present-biased parameters β and the beliefs about the future present-biased parameter . Principals obtain more expected profits from time-consistency preference agents than time-inconsistency preference agents and obtain more expected profits from sophisticated agent than partially naive and pessimistic agent. However, they obtain equal expected profits from sophisticated agent and partially naive and optimistic agent.

Key words: time-inconsistency preference, principal-agent problem, quasi-hyperbolic discounting, sophisticated agent, partially naive agent